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NIST Special Publication 789

Proceedings of the International Symposium on

Correlation and Polarization in Electronic

and Atomic Collisions

Paul A. Neill, Kurt H. Becker, Michael H. Kelley, Editors
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NIST Special Publication 789

Proceedings of the International Symposium on

Correlation and Polarization in Electronic

and Atomic Collisions

Held August 2-4, 1989 at Stevens Institute of Technology, Hoboken, NJ

Edited by

Paul A. NeUl

Department of Physics

University of Nevada

Reno, Nevada

Kurt H. Becker

Department of Physics

City College of New York

New York, NY

Michael H. Kelley

Electron and Optical Physics Division

National Institute of Standards and Technology

Gaithersburg, MD

June 1990

U.S. Department of Commerce

Robert A. Mosbacher, Secretary

National Institute of Standards and Technology

John W. Lyons, Director
National Institute of Standards U.S. Government Printing Office

and Technology Washington: 1990

Special Publication 789

Natl. Inst. Stand. Technol.

Spec. Publ. 789

163 pages (June 1990)

CODEN: NSPUE2

For sale by the Superintendent

of Documents

U.S. Government Printing Office

Washington, DC 20402
iii

Preface

The International Symposium on Correlation and Polarization in Electronic and

Atomic Collisions was held at the Stevens Institute of Technology, Hoboken, NJ on August 2-4, 1989. A gathering together of colleagues is always a pleasurable occasion

and it was a particular pleasure for Kurt Becker, Mike Kelley and myself to organize

this Symposium.

The previous meeting, held in Belfast, UK (1987), extended the scope of the

Symposium to include ion-atom and atom-atom collisions. This was reflected in the

Symposium title which was modified to replace the term "electron-atom collisions"

with "electronic and atomic collisions." The Hoboken meeting maintained this ex- panded subject area and included several talks in more loosely related areas. The

attendance of 66 scientists represented 16 countries and the talks described the work

of 23 research groups from 9 countries.

The opening address was given by Professor Joachim Kessler (University of Munster) . This was followed by 23 invited talks and closing remarks by Professor Benjamin

Bederson (New York University).

We would like to thank the speakers for giving permission to publish their con- tributions to the meeting. A special thanks to those who reformatted their text at my request. The organizers wish to acknowledge the sponsorship of the American

Physical Society and financial support from the U.S. National Science Foundation,

the National Institute of Standards and Technology and the U.S. Department of Energy, Office of Energy Research, Division of Chemical Sciences. We would also like

to thank the people of the Office of Center Operations at the Stevens Institute of

Technology.

Finally we wish to express our regret at the loss of our colleague Klaus Jost. Klaus was a talented scientist with a warm personality and will be missed by the community.

We dedicate these proceedings to his memory.

Paul Neill

University of Nevada, Reno

March 1990

In the same way that I am often referred to as the "head" of the group, Klaus was

the "heart."

J. Kessler, August 3, 1989

In Memory of Klaus Jost

The Proceedings of this Symposium are dedicated to Dr. Klaus Jost who died of a heart

attack in Miinster, Germany, on 15 July 1989 at age of 52. Klaus Jost had been in the

field of electron-atom collisions since 1959, when he joined my group at Karlsruhe as my first graduate student. His achievements and his advice were invaluable for the

success of this group in the following 30 years. Not only did the results of his Ph.D.

thesis become well known and were frequently reproduced (e.g., in Massey-Burhop,

Electronic and Ionic Impact Phenomena, Vol. I, p. 356 and 357, or the front page of

Physics Today, Nov. 1969), but his later achievements, like a convenient type of electrostatic spectrometer or very careful and reliable absolute cross-section measurements, were also highly appreciated by workers in the field. The latter work was done after he

had accompanied me to Miinster in 1971 where he belonged to the small number of

people who built up from scratch an experimental atomic physics group. He assumed

increasing responsibility for the management of this group and played a central role in

it. When we had the Symposium on Polarization and Correlation in Electron- Atom

Collisions in 1983 in Miinster he was one of the three organizers.

One of the reasons for my high respect of Jost is that his main object was not his personal career, but rather to do very good physics and to help others to do so. Although

he published a number of fine papers he never succumbed to the publish- or -perish

ideology, and instead of publishing piecemeal papers he always waited until all the

results were really ripe. His pleasant personality and his helpfulness made him very

popular both with students and colleagues.

The loss of Jost is a terrible shock for my group — it is impossible to replace him. The

reaction of the participants of this Symposium showed the high reputation he had

among his peers. I am very grateful that I had the privilege of having Klaus Jost's company for a long period of my life.

(Joachim Kessler)
VI

Disclaimer

Certain commercial equipment, instruments, or materials are identified in individual contributions to these proceedings in order to adequately specify the experimental

procedures. Such identification does not imply recommendation or endorsement by

the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
Contents

Preface -

In Memory of Klaus Jost '. A Choice of Open Problems in Polarisation and

Correlation in Electronic and Atomic Collisions

J. Kessler

Extracting Dynamics from Collision-Induced Polarization ^ U. Fano

Correlations in (e, 2e) Experiments. Recent Results on Inner and

Outer Shells

G. Stefani, L. Avaldi & R. Camilloni

Theoretical Atomic Physics Code Development at Los Alamos

R.E.H. Clark & J. Abdullah, Jr.

Systematic Investigations of Mott Polarimetry with Very Thin Gold

Films

M.A. Khakoo, J. A. Brand, W.M.K.P. Wijayaratna & T.J. Gay

Polarization Correlation Effects in the Two-Photon Decay of

Metastable Atomic Deuterium • A. Duncan

Electron Impact Excitation of the n=3 States of Helium and 5s^Pi

State of Krypton

A. Crowe

Orientation and Alignment in Electron Collisions with Sodium

P.J.O. Teuhner, R.E. Scholten & G.F. Shen

Theoretical Interpretation of the Associative Ionization Reaction

between Laser Excited Sodium Atoms: Atomic and Molecular

Points of View

A. Henriet & F. Masnou-Seeuws

Detailed Study of Polarization Effects on the Associative Ionization

in Na(3p) - Na(3p) Collisions

H. A.J. Meijer

Excitation and Ionization of Atoms - What We Can Learn by llsing

Polarized Collision Partners

K. Bartschat
Vlll

Alignment and Orientation in Simply-Structured Atomic Collision 70

Complexes

R. Hippler

Is the "Classical" Double Scattering Less Accurate for Calibrating 76

Mott Polarimeters than Recently Proposed Methods?

A'. Jost (Delivered by A. Gellrich)

Low Energy Electron Scattering by Sodium 83

L. Viiskovic

The Effects of a Finite Scattering Volume on the Determination of 89

Electron Impact Coherence Parameters

P.W. Zetner, S. Trajmar & G. Csanak

Spin Dependence in Electron Impact Ionization of Atomic Hydrogen 95

M.S. Luhell, D.M. Crowe, X.Q. Gilo, F.C. Tang, A. Vasilakis,

J. Slevin & M. Eminyan

Progress on Spin Detectors and Spin-Polarized Electron Scattering 102

from Na at NIST

J.J. McClelland

Coherences between Autoionizing States of Different Excitation 110

Energies

H.G.M. Heideman

Stokes Parameter Measurements in the Heavy Rare Gases 115

J.W. McConkey, P.J.M. van der Burgt, J.J. Corr & P. Plessis

Spin-Polarized Electron Impact Ionization of Atoms 121

G. Baum, P. Freienstein, L. Frost, B. Granitza, W. Raith &

H. Steidl

"Spin-Orbit" Effects in Inelastic Electron- Atom Collisions: 128

Are Experiments in Conflict with Theoretical Models?

G.F. Hanne

Optical Analysis of Electron Impact Induced Excited State- Excited 134

State Transitions in Alkali Atoms

B.J. Stumpf

The Stepwise Excitation Coincidence Technique 141

M.C Standage, IV.R. MacGillivray, A.J. Murray & C.J. Webb

Summary of Symposium Proceedings — Some Impressions and 147

Observations

B. Bederson

List of Attendees 155
OPENING ADDRESS

1

A CHOICE OF OPEN PROBLEMS IN POLARISATION AND CORRELATION IN

ELECTRONIC AND ATOMIC COLLISIONS

Joachim Kessler

Physikalisches Institut, Universitat Miinster,

D-4400 Miinster, Wilhelm-Klemm-Str. 10, West Germany

I have been privileged to deliver the official opening address of what promises to become a very stimulating Conference, because the Organizing Committee — Drs. Kurt

Becker, Mike Kelley, and Paul Neill — was very successful in persuading a number of internationally renowned speakers to give a review of their field.

This symposium has become one of the traditional satellite meetings to the

ICPEAC which does not mean, of course, that the topics discussed here are traditional.

On the contrary, the Symposium is focused on novel information obtained by advanced

techniques such as coincidence methods and polarization experiments. Keywords of the

previous conferences were polarization (of photons and electrons), correlation (of atomic-collision products), coherence, alignment, and orientation. But when you compare the

titles of the previous symposia, you find that they differ a little bit from each other, which

indicates that the emphasis has been varied. The terms "correlation" and "polarization"

have changed places from time to time, and at the Belfast meeting 1987 the "electron-atom" collisions have been substituted by "electronic and atomic" collisions. This

indicates that at Belfast the scope of the program was extended to include ion-atom and

atom-atom collisions which has been maintained at the present conference.

Such slight shifts of the emphasis may partly help to answer a question which was

occasionally discussed by some of the active participants. Does it really make sense to have such a conference every two years? Needless to say, 100 people or so produce a lot of

new data within a period of two years. But couldn't these be presented at the ICPEAC

just as well? Is there really a good reason for an extra symposium not only from time to time but every two years? Or is the main reason that such meetings tend to develop their

own dynamics and that once they have been started they jump into a stable rhythm?

Shifting the emphasis of the symposium makes it certainly easier to find a positive answer

to this question. ,

Still, I do not think that a positive answer can be taken for granted. It depends
2

very much on what one expects from a conference of this type and this depends on the

individual viewpoint. I can only give my personal view. I always found it a great advantage of the Polarization and Correlation Symposium that it is the proper place for discussing open problems and controversial questions. In Miinster we have developed into a

rather big group over the years, and when it came to the question of what to present at the ICPEAC and what at this Symposium, then I felt that we should present clear-cut

results which speak for themselves at the ICPEAC. Results, however, which we would like

to discuss in great detail with close colleagues from other groups should be presented at the satellite meeting. Examples are results that differ from those of other groups or whose

interpretation is not clear.

The Polarization and Correlation Symposium has always been a forum for discussing such problematic or controversial issues. Let me briefly mention some points of that

type we dealt with in the past and highlight some of the problems we certainly have to discuss at the present conference. At the Miinster Symposium in 1983 S. Trajmar's group

presented an experiment on superelastic electron scattering from laser-excited barium

atoms where they found an unexpected asymmetry, i Despite careful systematic changes of

the parameters of the experiment, the asymmetry did not vanish. A broad discussion of

this controversial question was quite important, because it was not at all clear whether

this was a new effect which was as yet unexplained or, more likely, an unresolved experimental artifact which — and that is why it was important — might affect other similar

measurements as well. It seems that tomorrow we will get the final answer to this open

question.

Another debate was opened by M. Lubell at the Miinster Symposium, and it may

persist for a few more years. It concerns the accuracy of electron polarimetry by Mott

scattering. This point came up because of major advances in polarized-electron sources with the result that the accuracy of many experiments with polarized electrons is no longer limited by statistics but rather by the value of the electron polarization. As a conse- quence, the determination of the accuracy of an electron-polarization measurement has

assumed a much higher degree of importance than before, when the papers of different

groups varied considerably in the accuracy they claimed for their polarization measurements. Conservative uncertainties were ±5%, more optimistic groups claimed uncertainties down to ±1%. Since I could never understand why the rules of Mott scattering for

polarization analysis should be so different in some other groups than they are in Miinster

I was quite relieved to hear that also the careful experimental analysis by Fletcher, Gay,
3

and Lubell2 showed that, for basic reasons, the overall uncertainty of Mott polarimetry is

restricted to at least ±5%.

This is one of the reasons why the discussion of electron polarimetry has revived in

the past few years and why new methods for polarization analysis have been suggested. In Miinster we have focused much attention on this problem. We found that optical polarization analysis utilizing the polarization of helium impact radiation that is produced by

polarized electrons is a very accurate method of electron polarimetry. It has been suggested by T. Gay a few years ago and is discussed in a broader context in the contribution by

K. Jost. In Miinster we are very upset because we lost K. Jost, one of my most competent

co-workers, shortly before going to the ICPEAC. His talk will be presented by

A. Gellrich. Incidentally, it is interesting to note that the discussion of electron-polarization techniques now centers on the polarization detectors and that the discussion of polarized-electron sources which had revived after the discovery of the Fano effect and the

GaAs source has slowed down. This is reflected by three topics on polarization analyzers

and none on sources in our program. It seems that the GaAs source will be the standard

source technique for a longer period of time and that the hope to find materials even more

suitable than GaAs will not be fulfilled for a while. The Fano effect, on the other hand,

turned out to be a valuable tool for studying photoionization processes and is nowadays

rather a topic of synchrotron radiation conferences than of this symposium. That is an example of how things can develop in directions not anticipated at their beginning.

Let me continue my list of controversial problems with an example from the field

of electron-photon coincidences. I am not going to take up again the well-known problem

of angular-momentum transfer <L^> in electron-impact excitation. There were different

views on the interpretation of the angular dependence of <L^>; the semi-classical

interpretation originally suggested by Kohmoto and Fano^ and favored by many

experimentalists was opposed by several theorists. I think the debate has come to a certain conclusion, though I am not so sure whether it is really over. From the various open

problems of electron-photon coincidences I would like to point out one, in which we are involved ourselves.

If you excite an atomic state by electron impact then you have certain restrictions

due to reflection symmetry. As a consequence, the waist of the charge cloud of an excited

P state must have zero height, 4 i.e., it does not look like in Fig. lb, but the waist goes

down to the origin of the coordinate system like in Fig. la, if there are no spin- dependent

forces that flip the electron spins in the excitation process. If you do have spin flips, then
4

Fig. 1. Collisionally induced charge cloud in P-state atom; (a) without spin flip,

(b) with spin flip. (From ref. 4)

reflection symmetry still holds for the whole system electron plus atom, while the individual wave functions will change their symmetry character. In this case the height of the

waist becomes different from zero corresponding to an oscillator along the z aixis, so that

light emitted in the y direction has not ordy a linear polarization component along the x

axis but also an incohent component along z. In other words the corresponding linear light

polarization, usually denoted by P4, is smaller than 1. An equivalent way to express this

is to say that the density-matrix element poo, which corresponds to excitation of the sublevel with the magnetic quantum number M = 0, is no longer zero. Briefly, poo 0 or P4 < 1 indicate spin flips, i.e., the significance of spin-dependent interactions. That is why electron-photon coincidences are one way of studying spin-dependent interactions in

electron- atom collisions.

Another more direct way are experiments with polarized electrons where the spin

flips can be directly observed by electron- polarization measurements. The point I want to make here is that there are some controversial results of the two methods. From the electron polarization experiments made over the past 2 decades one knows quite well where

spin-dependent interactions do or do not play a role. One knows, for instance, that spin

interactions of the scattered electrons have negligible effects for small scattering angles at medium energies This is in accordance with simple models and also with sophisticated

calculations. On the other hand, there are recent electron-photon coincidence measure- ments by several groups yielding significant values of poo, i e. significant spin interaction.
5

at such angles and energies. This is certainly puzzling and the question of artifacts is raised again. Since in Miinster we are involved in both electron polarization and electron-photon coincidence measurements, G.F. Hanne will try to shed some light upon this

dark point, and I think that also J.W. McConkey will address this problem.

From the examples given it might appear that puzzling problems occur only in

involved processes like coincidence experiments or polarization phenomena. But this is not

at all true. Let me quote from a paper by Callaway and McDowell^ saying: "One purpose

of this Comment is to remind our colleagues, both theorists and experimentalists, that

while it is of great interest to measure or calculate such esoteric parameters as spin-flip

ratios, differential cross sections and orientation and alignment parameters, we are still unable to give a satisfactory answer to the simple question 'What is the absolute value of

the cross section (for electron impact excitation of a specific energy level in atomic hydrogen) at a specified energy?' We find that the uncertainties are of the order of 20% at some

energies for the simplest cases, of at least a factor of two in other apparently simple

cases..." . :

Somewhat later I discussed in a Comment^ why in a perfect scattering experiment,

which yields the complete set of observables describing the collision process, it is usually

the cross sections that are the weakest point. At least in those investigations that have

been done with heavy atoms in the past few years it was clearly the cross sections that

had the largest experimental and theoretical uncertainties. A similar problem in which the

Adelaide group is involved are serious discrepancies between experimental and theoretical

cross sections occuring with sodium but not with other atoms, even though the experimental and theoretical methods applied to the various targets are the same.

Given this situation, you hear the question between the lines I quoted before: does

it make sense that you study all those esoteric parameters as long as we have problems

with such fundamental parameters as cross sections? To me the answer becomes clear

when I look at the development of collision physics in the past 2 decades. When the first experimental results on electron-photon coincidences, (2,2e), or spin polarization were

published, most of them could not be interpreted quantitatively within the framework of

the existing theoretical approaches, which had been focused on the calculation of cross

sections. The novel, much more detailed experimental data were an immense challenge for

theory resulting in great advances of the theoretical methods. In my own field, for instance, it was impossible a decade ago to obtain reliable theoretical results for electron

collisions with heavy atoms where the different kinds of spin interaction simultaneously
6

play a role. Today one finds relativistic approaches including exchange and charge-cloud

polarization which yield reliable theoretical data for quite complex processes of which

K. Bartschat will give some examples tomorrow. The wealth of the so-called esoteric ex- perimental data put theories to a very stringent quantitative test so that only the most

reliable of them can survive. And these give usually a better description of cross sections

as well, even in cases where this is not necessarily anticipated. An example for this was

given by McEachran and Stauffer^ at the previous symposium, where they discussed cross

sections for electron scattering from heavy atoms at low energies, where one has Ramsauer

minima below 1 eV. They showed that even at these low energies a relativistic theory

based on the Dirac equation gives much better results for the Ramsauer minima than

non-relativistic approaches. Generally speaking, I feel that the elaborate experimental

and theoretical methods which are discussed at our symposium enabled us to make a quantum jump in the quantitative understanding of collision processes.

The few examples, which I picked out according to my personal preference, show

that we will have more than enough problems to discuss here. It seems that our Australian friends feel that this will also be the case in 2 years because they are planning the

next symposium in 1991. I cannot foresee whether the group of people assembled here will

continue to produce a sufficient number of exciting problems to have another stimulating

symposium 4 years from now in Europe. Perhaps Ben Bederson can make a good guess

when he will close the symposium and when all of us will be much better informed than

we are today.

References

1. D.F. Register, S. Trajmar, G. Csanak, S.W. Jensen, M.A. Fineman and R.T. Poe,

Phys. Rev. A 28, 151 (1983)

2. G.D. Fletcher, T.J. Gay and M.S. Lubell, Phys. Rev. A 34, 911 (1986)

3. M. Kohmoto and U. Fano, J. Phys. B 14, L 447 (1981)

4. N. Andersen, J.W. Gallagher and I.V. Herte), Physics Reports 165, 1 (1988)

5 J. Callaway and M.R.C. McDowell, Comments At. Mol. Phys. 13, 19 (1983)

6 J. Kessler, Comments At. Mol. Phys. 18, 279 (1986)

7. R.P. McEachran and A.D. Stauffer, in "Correlation and Polarization in Electronic

and Atomic Collisions" (A. Crowe and M.R.H. Rudge eds.) p. 183. World Scientific,

Singapore 1988
7

EXTRACTING DYNAMICS

FROM COLLISION-INDUCED POLARIZATION

U. Fano

Dept. of Physics & James Franck Inst.

; • University of Chicago, Chicago, IL 60637

1. INTRODUCTION

The polarization generated by collision depends on the dynamics of the collision process

but also on incidental aspects of the experimental designs. Identification of the relevant

dynamics and evaluation of its essential parameters require one to disentangle these parameters from raw collision data, a task common to many theoretical studies.^ C.W. Lee's report

to the 4th Symposium^'^ has contributed many important elements of the polarization dynamics, but has still fallen short of the actual extraction of dynamical parameters. Intervening

progress toward Lee's goal is reported here.

The essential parameters of interest are elements of the scattering (or equivalently of the

transition or the reaction) matrix, preferably freed from magnetic quantum numbers — and

thus made rotationally invariant ~ through the Wigner-Eckart theorem. Polarization and cross

section measurements are represented by real numbers and expressed as quadratic — more

properly, bilinear ~ functions of the dynamical parameters, among which the scattering and

transition matrix elements are complex. Moreover the measurements are obtained at sets of

scattering angles, whereas the rows and columns of dynamical matrices are labeled by angular

momentum quantum numbers. Disentangling dynamical parameters requires thus some combination of expansion into spherical harmonics with unraveling of bilinear equations. It seems
8

likely at this time that this unraveling should precede the harmonic expansion.

The initial step appears trivial for the prototype collision

e+HQ(ls^)-^e'+Hc{\s2p), (1)

where cross section and orientation data are familiarly translated into a pair of real parameters

{^(9),X(0)},'^ linerization is obtained here simply be using VX and ^l-X instead of A,. A full

disentangling (or "inversion") procedure is otherwise nontrivial; its development through Lee's

and later contributions remains fragmentary. Lee's papers have since been extended to alignment parameters of process (1) ^ but this advance remains peripheral to the main task because

orientation and ahgnment measurements bear on the same parameters {'k,x} in this case.

2. SUBTRACTION OF LONG RANGE CONTRIBUTIONS

Long range interactions between projectile and target contribute heavily to cross sections

and polarizations for charged-particle collisions. On the other hand their contribution through

each partial wave component with large orbital momentum remains small and can be

evaluated accurately in the Born approximation. Moreover the aggregate Bom contribution of

all partial waves is represented compactly in terms of the "form factor", i.e., of the Fourier

transform of the charge distribution of a target's excitation of interest; this parameter is

obtained readily from theory or by measurements on collisions by very fast charged particles.

Recalling that the contributions of long range interactions cause inelastic cross sections

to peak sharply at small scattering angles, one anticipates a slow convergence of their expansions in spherical harmonics. There results a requirement to consider large sets of dynamical

parameters. This substantial inconvenience can, however, be removed by two devices,

namely:
9

a) Replacement of the total angular momentum / of projectile + target, which ranges to

infinity, by the angular momentum transfer ji which does not exceed the sum of the target's

angular momenta in its initial and final states. This device, used throughout by Lee, ' also

limits the difference of the initial and final projectile orbital momenta but does not restrict

their magnitudes.

b) Evaluation of the dynamical parameters corresponding to large orbital momenta by

the Bom approximation, which is accurate in that range and is readily applied as noted above.

The relevant procedure^ is outlined below.

An analysis of the Bom approximation's contribution to the cross section and orientation

in the process (1) has been presented previously for purposes of illustration.^ Each of the

observed quantities was viewed in Ref. 3 (IV) as proportional to a squared element of transition matrix, |r(0)p, T itself being the sum of a Bom term and of a residual term .

Contributions to |r(0)p consist then of three components labeled as BB, RR and I(BR) ~

where I stand for "interference" - adding up to a "total". (The BB contribution to the orientation vanishes, of course, but I(BR) does not).

Ref. 6 proceeds in the same manner for the dynamical parameters whose indirect connection to experimental data is the subject of this report. Each of these parameters is cast as an

"exact" transition matrix element T^^ , labeled by rotationally invariant angular momentum

quantum numbers, and is viewed as the sum of two contributions

j^ez _ j'B _^'yR — Y^

-\-(J[^ —Y^

^ (2)

itself is constmcted in terms of the form factor for the relevant transitions of the collision

target, a quantity measured or calculated for collisions of very fast charged particles, that is,

without any allowance for projectile-target exchange interactions or other corrections
10

appropriate to lower energy transitions. Such corrections are thus incorporated in 7^, whose

convergence to zero with increasing orbital momenta is thereby hampered but still readily

tractable.

3. UNRAVELING BILINEAR EQUATIONS

An initial approach to determine the dynamical parameters of process (1) by adapting the

analytical machinery of Ref. 3, complemented by subtraction of the Bom contribution, has

been made by John Bohn in 1988.^ Its numerical implementation proved, however, unstable.

Bohn initiated then a more flexible procedure applicable to inelastic electron impact on any

target with zero angular momentum.^

Flexibility stems from unraveling initially a set of intermediate parameters, namely, the

probability amplitudes of excitation to specific states [/,m,), designated by {aj^^ iQ)} as in the

review by N. Andersen et al.^ The index of this amplitude can be omitted since it

represents the specific angular momentum of the excited target, its dependence on the scattering angle 6 is preserved in the unraveling process. The set of observable tensorial parameters

of the excited target, <T^'^^>j^Q, equals a bilinear expression in {a^ iQ), a^ '(Q)}. This

approach thus unravels the bilinear expression before expanding it into spherical harmonics, in

contrast to the Lee-Fano procedure of expanding the <T^'^^>q directly.''

Bohn noted that the subset of parameters with cylindrical symmetry about the collision

axis, {<r^^^>^^0} consists of jj+l nonzero elements with {k=G,2,...,2ji), whose values are

linear combination of {j«^^(6)p} with {m,=0,l,...,y, ). [Negative values of are not independent owing to reflection symmetry through the scattering plane]. The amplitude moduli

Ifl^ (9)1 are thus determined from the observables <T^''^>j^q by solving a linear system of
11

inhomogeneous equations. ' t.- ••. i: .

The phase differences among the {a^^{Q)} are to be determined next from the observables {T^'^^}j^Q, with nonzero q=-k, -k+l, • • • -1,1,2, k, each of them represented as a

superposition of

«m,(6)<+^(e) = KIK+Jexp/(x^/e)-x^,.,^(0)). o)

Measurement of the <T^'^^>j^Q parameters suffices in fact to determine the sines and cosines of

all the phase differences. [The remaining "arbitrary" phase of the [a^^] reflects a convention

on the states [/fm,)]. ' >.: -.. rJ'^'

The desired invariant dynamical parameters, indicated in Ref 3 (III) by (L^Lg\S (j[)\lalh),

where L^=0 and LQ=ji in our case, are finally constructed from the coefficients of the expansions of the a^ (Q) into associated Legendre functions Pj ,^ {Q). Specifically they are given by

Eqs. (2) and (4) of Kohmoto and Fano^ as linear functions of these coefficients and of Wigner

coefficients for additions of angular momentum vectors 7^=7^+7^, L^=Z^+7^.

Extensions of this procedure to targets with nonzero angular momentum, to account for

electron spin effects and to deal with projectiles other than electrons, remain to be developed.

The following should, however, be anticipated with regard to asymmetric targets with nonzero

angular momentum. Determination of the full set of relevant dynamical parameters implies

maximal experimental knowledge of the collision variables, including the target's polarization

before as well as after the collision. (The initial polarization can be controlled by appropriate

preparation of the target). Less-than-maximal knowledge of the target's <T^'^^> (or of projectile spin polarization) implies in turn a reduction in the number of equations available to

determine the dynamical parameters.
12

Preliminary Calculations

Practical aspects of the procedure outlined in this Section have been explored in recent

weeks by A. Chakravorty and X.C. Pan, utilizing data on process (1) from various sources to

evaluate the dynamical parameters (L^ Lg |S (/, h )• [Th® S and T matrices coincide here for

inelastic processes]. Interpolation and extrapolation of data from different authors and at

different angles proved necessary to provide a consistent set of input data. The procedure

appears workable but considerable experience and skill may prove essential to its sound and

effective application.

A particular difficulty emerged in the selection of matrix elements. values

obtained from form factors yield a cross section at 0=0*^ about 40% in excess of the experimental value at 80eV. Scaling them down to match experiment at 0=0'' distorts the fitting of

S(Jt), since the departure from Bom at 80eV depends on the angle 0. Convergence of

jex_jB j^^g ^j^yg required the fitting of >9 harmonics; dispensing with scaling, on the other

hand, avoided a major kink in the dependence of S (/, ) on , but may slow down its convergence. It remains obscure which among these or other versions of the procedure may lead to

a more substantive interpretation of experimental evidence. On the other hand the very raising of specific issues appears to open new vistas.
13

<LsL^\S(J,ilJ,> = ]^\e^^

(Ls=hL^ =0,7- =1)

Unsealed Born

^\ (a.u.) (|) (rad)

la

IS| (a.u.) <{) (rad

0 0 0.12 0.07

1 0.11 0 1 0.18 1.07

At2 0 n 3.19 2 0 16 1 41

3 0.26 ' 3.65 3 0.13 1.25

4 0.26 3.68 4 0.11 1.23

5 0.22 .. . 3.66 , . 5 0.09 \ 1.40

6 0.16 3.75 6 0.07 1.71

7 0.11 3.92 7 0.05 1.91

8 0.08 . ^ 4.09 8 0.02 1.94

9 0.05 4.08 9 0.01 1.09

10 0.03 - 3.74 10 0.01 0.95

Scaled Born

(/fc=/^-l iS|(a.u.) (})(rad) (/fc=/l+l) lS| (a.u.) ({) (rad)

1 0.18 0

2 0.06 3.25

3 0.20 3.85

4 0.21 3.85

5 0.17 3.85

6 0.12 4.03

7 0.09 4.30

8 0.07 4.60

9 0.04 4.76

10 0.02 4.78

0 0.17 0.05

1 0.22 0.80

0.18 1.06

3 0.16 0.90

% - ' 0.13 0.87

5 0.10 0.99

6 0.07 1.19

7 0.05 1.18

8 0.03 , 0.70

9 0.03 0,29

10 0.02 0.38

Remarks:

1) Smooth convergence of 15 1 as increases

2) Sensitivity of ^ to precision of input.
14

4. INTERPRETATION OF THE DYNAMICS

The extraction of dynamical parameters discussed here aims at providing a transparent

interpretation of the observed polarizations. Qualitative aspects of the polarization, e.g., the

induced target orientation being perpendicular to the scattering plane and the alignment tensor

being symmetric under reflection through that plane, follow from symmetry considerations.^

More detailed aspects, typically the sign of the orientation, depend on dynamical features that

have not yet been identified fully.

Fragmentary interpretations of the orientation sign have dealt thus far mainly with semiclassical aspects of collision, such as the curvature of the projectile's trajectory. On the other

hand quantum mechanical interpretations would generally center on the analysis of transitions

between (time dependent) stationary states. Some relevant aspects of the dynamical parameters {L^LQ\S(ji)\lalt,), namely, identification of their bilinear combinations that are real or

purely imaginary, as well as even or odd under a certain permutation of their indices, have

been noted in Ref. 3 (III). This permutation bears on the dynamical propensity toward a

decrease of the orbital momentum about the center of motion of projectile and target accompanying target excitation at the expense of the projectile's kinetic energy. This kind of

analysis looks promising but has not yet been pursued to sufficient depth.

ACKNOWLEDGEMENT

The work leading to this report has been supported by NSF Grant PHY-86-10129 and

includes contributions by John Bohn, Alak Chakravorty and Xiao-Chuan Pan.
15

REFERENCES - :

' - ;

1. A prototype extraction of basic parameters from experimental data is seen in the

representation of Rydberg series of spectral levels through its limit and its quantum

defect, parameters whose accuracy appears more significant than that of individual levels.

Additional examples are stressed in U. Fane and A.R.P. Rau, Atomic Collisions and

5pecrra (Academic, Orlando 1986).

"

2. C.W. Lee, in 4th Intern' I Symp. on Polarization and Correlations in Electronic and

Atomic Collisions,

3. C.W. Lee et al., Phys. Rev. A: I 33, 921 (1986); II 34. 959 (1986); m 36. 66 (1987); IV

36. 74 (1987); V 39, 5 (1989).

4. See, e.g., N. Andersen et al. Physics Reports 165. 1 (1988).

5. R. Venugapolan, B.S. Honors Thesis, U. of Chicago (1987), unpublished.

6. X.C. Pan and U. Fano, Phys. Rev. 39, 4502 (1989).

7. J. Bohn, B.S. Honors Thesis, U. of Chicago (1988), unpublished.

8. J. Bohn, to be published

9. M. Kohmoto and U. Fano, J. Phys. B. 14, L447 (1981).

10. "Maximal knowledge" of tensorial parameters <T^'^^> may be provided indirectly, e.g.,

by magnetic polarization of a target in a thermal bath. In this case statistical mechanics

determines the nonzero values of the <T^''^> with q=Q corresponding to axial symmetry

about the field direction; this symmetry causes parameters with q ^0 to vanish under conditions of thermal equilibrium [see, e.g., U. Fano, Rev. Mod. Phys. 29, 74 (1957) sec.

4g]. ,y
16

CORRELATIONS IN (e,2e) EXPERIMENTS. RECENT RESULTS ON INNER AND OUTER SHELLS

G. Stefani

Dipartimento di Hateraatica e Fisica. Universita' di Camerino, 62032 Camerino, Italy

L. Avaldi and R. Caroilloni

Istituto di .Metodologie Avanzate Inorganiche CNR

Area della Ricerca di Roma. CP 10 00016 Monterotondo , Italy

Introduction

Among the electron impact ionisation

experiments the (e,2e) experiments measure

triple differential cross sections (TDCS),

in as much as they consist of measuring

simultaneously the energy E^ of the incident electron, the energies E^ and of

the two escaping electrons and the probability that these coincidence pairs are emitted into solid angles around the directions i^^, </'a} srid iS^,<^)^ The electronelectron coincidence technique has been

largely applied to obtain structural data

on atoms and molecules and to perform

studies on the collision d^mamics. The

technique also turned out to be a good

mean to investigate ee-correlations . Two

classes of ee-correlations can be reco- gnized to be relevant in a (e,2e) process:

i) ee-correlations present in the isolated

atom or molecule under consideration, in

the initial ground state or/and in the

final ionic state. They manifest themselves mainly as satellite lines in the coincidence energy spectrum; ii) ee-correlations that- are switched on during the

ionising collision. They involve the free

electrons, interacting with each other or with the bound atomic electrons. Inclusion of these latter correlations has

proved in a few cases to be essential in

calculating the correct shapes of the

angular distributions and the size of the cross section. Depending on the kinematics of the experiment, the structural

aspect of the information can be privileged with respect to the d^mamical one, or viceversa. Most of the work done to

investigate electronic correlations has

been concentrated on outer shells of atoms

and molecules, studying the satellite

peaks and their angular distributions in

symmetric kinematics (E^=Ej_^). In these

experiments the high energy incident electron undergoes a high momentum transfer

(K>. collision, thus essentially interacting with a single target electron, remo- ving it and leaving the residual ion in a given quantum state (impulsive ionisation).^ Under these conditions are the

initial and final state correlations ( ISC

and FSC respectively ) of kind i) to be

investigated. On the other hand, high/medium energy and low momentum transfer

scattering (E^>5 I. P., E^»E^ ) must involve grazing collisions, with the incident

electron interacting with the target system as a whole. These latter kinematics

are more suitable for dynamical studies.

Asymmetric (e,2e) experiments were recen- tly carried out on both valence and innervalence atomic orbitals. Post Collisional

Interactions ( PCI ) and ContinuujD Final

State Correlations ( CFSC ) were evidentiated by these investigations .'^ j ^ The present paper will be confined to

report on some recent experiments performed in asymmetric kinematics (incident

energy larger than 1 KeV) aimed at investigating ionisation d)mamics and ee-corre- lation effects for either cere and cuter

electrons

.

Inner shell TDCS

There is interest in measuring the

TDCS for inner shell ionisation because it

is expected to produce detailed information on the many-body response of an atomic or molecular system to the change of

the potential when the ionised system

relaxes. Until now the smallness of the cross section and the limitations imposed

by the "unavoidable" accidental coincidences have hampered extensive studies on inner shells to be carried out.

Here, the measurements of the TDCS for

the ionisation of C <5'ls in the molecule

C^H^ are reported. The (e,2e) experiments were performed in asymmetric conditions at

small scattering angle, with a scattered
17

electron energy (E^J of 1500 eV, and low

ejected electron energies ( 9.6 and 41.0

eV) . ^ In particular the (e,2e) technique

was applied for the first time to the

ionisation of an inner orbital upon kinematics where resonant channels are expected to be relevant with respect to the

direct ionisation.

Several dynamical effects were observed in the measured TDCS . The angular

distribution, reported in Fig. 1, shows a large recoil lobe and severe deviation of

the symmetry axis of both recoil and binary peak maxima from the momentum transfer

direction. The disagreement with 1st Born

predictions is more pronounced than for

the outer shell ionisation performed in

similar kinematic conditions. In particular, the recoil intensity is shifted to

form smaller angles with the incident beam

direction. This latter finding is in

contrast with what is usually observed^

Finally, the transition energy of the

(e,2e) peak is found to be shifted, for

the first time, with respect to the corre- spondent XPS transition.^ The amplitude of

the shift decreases with increasing the

slow electron energy as shown by the insert of Fig. 2. The angular distribution of the TDCS

is not accounted by a dipolar approximation which, instead, would account for the

equal intensity of the two opposite lobes

and which was found to be adequate in

describing the DDCS for similar molecules

upon similar kinematics.^ On the basis of

the results of the extensive theoretical

work done for the outer shell ionisation

and because of the localisation of the core orbital it is argued that the description of the free electrons wave functions

in the region of small r is crucial to the

description of the process (Distorted Wa- ves and approximations beyond Ist-order).

Furthermore, a suitable model for

inner-shell ionisation should take into

account the rearrangement of the residual

ion, that for light atoms ends up in the

emission of Auger electrons. When the

primary ionisation event is close to threshold, the ejected electron, slowly rece- ding in the ion field, and the fast Auger

electron can exchange energy and momentum.

As a consequence of this kind of post

collision interaction (PCI), a shift of

the (e,2e) peak and distortion of the

angular distribution is expected. A serai

classical model ''predicts an energy exchange from the slow electron to the Auger

insufficient to account for the "apparent"

shift of the (e,2e) peak in the energy

separation spectrum. It amounts to 0.46 ±. 0.23 eV at 9.6 eV ejected energy. A tentative explanation for the extra contribution asks for a non Frank-Condon unresolved vibrational structure shifting the

centroid of the peak. The vibrational

excitation in core level ionisation accom- panies the spatial redistribution of the

outer electronic charge. The possibility

of a non Frank-Condon distribution is

based on the following observations: i)

the C Is ionisation, at the energies

above threshold of our experiment, proceeds through resonant (5"^ls-*Kjj<S'u. channels that are expected to take into ac^- count up to 30% of the total intensity;

ii) strong non Franck-Condon effects have

been already observed in the photoionisation of the valence IVT'^^ orbital whenever

a resonant channel becomes dominant in the

ionisation process.^ The presence of such

channels would also contribute to the

distortion observed in the TDCS angular

distributions

Correlations and satellite structures

The satellite peaks present in the

ionization of the valence orbitals provide

evidence for the presence of ee-correlations in the target. The Ar 3s ionisation

spectrum can be considered a prototype

one, because it is characterized by a rich

satellite structure due to correlations in

both the initial and the final states, and

because it has been extensively investigated by photoionisation and impulsive

(e,2e) experiments. He is the other limiting case, where the correlations are present only in the initial state.

The (e,2e) experiments on these systems, that are reported in this paper,

have been performed upon asymmetric kinematics, intermediate between the impulsive

and dipolar regimes. Aim of this work is

to study the satellite intensity as a function of the momentum transfer (K) in

the collision.

Argon

The satellite spectrum from 35 eV up

and above the Ar ionisation threshold

has been measured at an incident energy of

1 Kev and at a fixed ejected electron
18

energy of 120 eV. The measurements have

been performed at three different values

of the scattering angle, in order to span

K values from 1.3 a.u. up to the Bethe

ridge value of 3.0 a.u., where the collisions are known to belong to the binary

regime. The satellite spectrum shows several unresolved broad structures and, by

using a model function deduced from the

main 3p~ line, the spectrum was deconvoluted into three main peaks . The relative

intensities, obtained as best fit parameters, are reported in Fig. 3 versus (a)

the recoil momentum, q=Ko- ( K(i+Kjj ) and (b)

the momentum transfer K. Previous symmetric (e,2e) results are also shown in the

figure. For the most intense structure,

whose centroid is at 38.5 eV, the relative

intensity is found to be independent on

the ionization kinematics. This structure

is known to be dominated by a transition

belonging to the manifold. The discrepancy between the intensity measured by

(e,2e) and XPS experiments, already observed in previous work, is confirmed.

On the contrary, the intensities of

the peaks centered at 36.5 and 41.3 eV.

show a monotonic dependence on K. This

behaviour is explained with a sizeable

contribution from transitions belonging to

the *D and manifolds, which are not due

to final state correlations. It has to be

noted that previous symmetric (e,2e) experiments assigned these peaks to *S states.

Because states with and ^P final configurations can be accessed not only due to

initial state correlations. but also as a consequence of continuum final state cor- relations, their excitation probability

may depend on the kinematics of the collision. The presence of such contributions

results also from a recent symmetric

(e,2e) experiment. In this latter study a dependence on the recoil momentum, q, of

the relative intensity of the satellite

structures is also evidentiated

.

Helium

The ionisation of helium to the n=2

excited state of He''' has been studied by

measuring the angular dependence of the

TDCS , for both the binary and the recoil

lobe, and comparing the shapes of the cross section with those relative to the

final ground state He* n=l . The asymmetric kinematics are characterized by high

energy, £^=1544.5 eV for n=l and £^=1585.4

eV for n=2 , small momentum transfer, K^O .

8

a.u., corresponding to 4 for the scattering angle and 20 eV for the kinetic ener- gy of the ejected electron. In these

kinematics the collisional regime is out

of the Bethe ridge and approaches the

dipolar one. The transition energy

spectrum is investigated in a region free

from resonances. The TDCS is shown in

Fig. 3. The most striking feature is the

intense recoil lobe present in the n=2

TDCS. Furthermore, while the angular

distribution for the n=l state is well

symmetric around the K direction, the

binary and recoil lobes for the n=2 state

deviate from the K symmetry by 10 and

25° respectively. Because the velocities

of the outgoing electrons are exactly the same in the two cases, the deviation can not be ascribed to the effect of long

range repulsive interaction between the

electrons. What is to be understood is

why the interactions of the free incoming

and outcoming electrons with the residual

core (nucleus+electron) are enhanced when

the ion is left in the excited state.

The transition to the n=2 final state is

made possible because of the presence of

ee -correlations in the ground state of

helium. The n=2 momentuni distribution

measured by previous symmetric (e,2e)

experiments,'!^ indicates an average <q>

value larger than the one associated to

the n=l final state. Larger <q> usually

means lower <r>, hence a higher probability for large angle elastic scattering on

the nucleus can be expected.

The ratio (3^ (n=2 )/g (n=l ) for the recoil

lobes has been reported as a function of

the recoil momentum q measured in the

experiment. At the lower q the values of

the ratio are slightly smaller than those

measured by symmetric (e,2e) experiments,

while with increasing q a much steeper

rise is observed. This different behaviour clearly indicates that, although the

width of the angular distribution of ejected electrons is basically determined by

the electron momentum density of the initial state, the momentum balance q measured in asymmetric conditions can not be

identified with the momentum exchanged by

the bound electron, unless Bethe ridge

conditions are fulfilled. In general this

q balance is a collective quantity and it

is influenced by the overall distortion of

the electron wavefunctions. *
19

Figure 1. The (e,2e) angular distribution for the ionisation of C ©"Is in C^^H, versus 5^, • Eo=1801.2 eV, E^=9.6 eV, 9'a.=4*', K=1.26 a.u.; (b) Ea=1832.4 eV,'

Eb=41.0 eV, 9^=5°and K=1.46 a.u. The energy Eo. is 1500.0 eV in both the

figures. Full lines: 1st Born model [formula (3) from Ref. 14].

Figure 2. The (e,2e) energy separation spectrum. Kinematics as in Fig. 1(a),

fixed at 60". In the insert the centroids of the s'ls peak (•) are reported versus E^, together with the Is binding energy (») measured by XPS
20

Figure 3. Relative intensity of the three main features of the Ar 3s satellite

spectrum plotted versus q (a) and K (b). The transition Ar 3s~"* is assumed to

be 100. Experimental data : (») from Ref. 10, (o) Ref. 15, (a) Ref. 16,

(#) Ref. 17. Crosses: theoretical predictions, Ref. 18. Arrow: XPS value.

Straight lines: least-squares fits to the data.

Figure 4. The (e,2e) angular distribution in helium, versus e^^ with

Eq^-1580 eV, E^=1500 eV, ©^=4 ° , K=0.8 a.u.. In (a) the final state

is He"^ n=l, in (b) n=2. The dashed lines are fitting curves to the

data.
21

References

1. H. Ehrhardt, G. Knoth, P. Schlenmier

and K. Jung, Z. Phys.D, 3 (1986).

2. I.E. McCarthy and E. Ueigold, Rep.

Progr. Phys.51, 288 (1988).

3. G. Stefani, Proceedings of the XV

International Conference on the Physics of Electronic and Atomic Collisions, Brighton, United Kingdom, 1987,

edited by H,B. Gilbody, W.R. Nevell

F.H. Read and A.C.H. Smith (North Holland, Amsterdam, 1988), p. 163 and re- ferences therein quoted.

4. L. Avaldi, R. Camilloni and G. Stefani,

submitted to Phys. Rev, A (1989)

and Book of Abstracts of Third European

Conference on Atomic and Molecular Physics, Bordeaux 1989, Editor A. Salin,

(published by EPS).

5. R.G. Cavell,J. Electron Spectrosc. Re-

. lat. Phenom. 6, 281 (1975).

6. P. Letardi, R. Camilloni and G. Stefani

Phys. Rev. B, (1989), to be published.

7. R.B. Barker and H.W. Berry, Phys. Rev.

51, 14 (1966).

8. D. Lynch, M.T. Lee, R.R. Lucchese and

V. McKoy, J. Chem. Phys. 80, 1907 (1984).

9. J.L. Dehmer, A.C. Parr and S.H. Southworth, in Synchrotron Radiation Handbook, Vol. 2, edited by G.V. Marr

(North Holland, Amsterdam, 1987).

10. L. Avaldi, R. Camilloni and G. Stefani,

Phys. Rev. A, (1989) to be published.

11. I.E. McCarthy, R. Pascual , P. Storer

and E. Weigold, to be published.

12. J.P.D. Cook, I.E. McCarthy, A.T. Stelbovics and E. Weigold, J. Phys. B17,

2339 (1984).

13. A.D. Smith, M.A. Coplan, D.J. Chornay,

J.H. Moore, J. A. Tossell, J. Mrozek,

V.H. Smith and N.S. Chant, J. Phys.

B19 , 969 (1986).

14. Yu.V. Popov and E.K. Shabalina, J. Phys. B19, L855 (1986).

15. K.T. Leung and C.E. Brion, Chem. Phys.

82, 87 (1983).

16. I.E. McCarthy and E. Weigold, Phys.

Rev. A31, 160 (1985).

17. J.F. Williams, J. Phys. Bll, 2015

(1978).

18. I.E. McCarthy, private comunication

1988.
22

THEORETICAL ATOMIC PHYSICS CODE DEVELOPMENT

AT LOS ALAxMOS

R. E. H. Clark and J. Abdallah, Jr.

LOS ALAMOS NATIONAL LABORATORY

Recently, several computer codes for calculation of various types of atomic physics data

have been developed at the Los Alamos National Laboratory. The purpose of the code

development was to provide a set of codes

that are very easy to use, will provide results

for any ion or atom, use the best possible

theory within the constraints of reasonable

running time, and provide great flexibility to

the user. The codes relevant to this meeting

are the CATS^ atomic structure code based

on Cowan's" Hartree-Fock method, the ACE^

collisional excitation code which can calculate collisional data in the distorted wave approximation (DWA) of Mann'^ or via first or- der many body theory (FOMBT), and the

TAPS^ code which can display various quantities from CATS and ACE and can provide

differential cross sections (DCS) and electron

impact coherence parameters (EICP).

For the three codes, CATS, ACE, and

TAPS, a brief description of the theory will be

presented. Following this will be a presenfation of the method of using the codes. Finally,

several examples of DCS and EICP calculations using various approximations available

in ACE will be given.

The atomic structure code, CATS, is based

on Cowan's^ Hartree-Fock method. Radial

wave functions are calculated in the single configuration Hartree-Fock approximation. Configuration interaction and intermediate coupling mixing is brought in through

perturbation theory. The calculation of the

radial wave functions is a non-relativistic cal- culation except that the mass and Darwin

terms can be included. Quantities calculated

by the CATS code include the radial wave

functions, energy levels, oscillator strengths

and, optionally, plane wave Born collision

strengths. The CATS code is run via com- mands. The following set of commands will

generate atomic structure data for neutral

barium:

opend bal

ion 56 1 rcf

6s2

6p2

6sl 6pl

6pl 5dl

/run end

All data calculated by CATS is placed on a random access file for easy retrieval by the

ACE and TAPS codes.

The collisional excitation code, ACE, reads

the atomic structure data from the random

access file prepared by the CATS code. The

radial wave functions for the target states can be generated in the Hartree-Fock or Hartree-statistical-exchange approximations

of Cowan^ with or without the inclusion of

the mass and Darwin terms. The ACE code
23

also contains subroutines to generate hydrogenic bound state wave functions for comparison purposes. The continuum wave functions can be calculated in the DWA method of

Mann or via FOMBT^'^ theory. Subroutines

are available to calculate Coulomb functions

for comparison purposes. A variety of local

exchange potentials are available. Unitarization of the reactance matrix can be turned on and off. Mixing in the target states is included. If requested, the scattering amplitude will be written to a file for use in

the TAPS code for DCS and EICP calculations. The ACE code is also run by commands. The following set of commands will

calculate cross-section data for neutral barium:

files bal

seq 4

-

setl 15

setle 1 5

0.0

2.24

eunits ev

e 20 50

Imax 250

dcs t thetaUn 37 0 180

go

end

All data from the ACE code is also written

to a random access file. The display code, TAPS, reads data generated from the CATS and ACE codes. It displays selected data in either text or graphic

form. Sums and averages over fine structure

levels and LS terms can be performed to provide data for LS terms and configurations.

The TAPS code can use the scattering amplitude generated by the ACE code to calculate

DCS's and EICP's. Various sets of EICP's

are written to different files which are named

Blimi, Fano, Hertel, and Stokes for obvious

reasons. The TAPS code is also rim by

commands. The following set of commands

will genetrate DCS's and EICP's for barium:

file adam

ion 56 1 4

dcs

cop

end

The following figure shows a schematic di- agram of the system of codes:

THEORETICAL ATOMIC PHYSICS CODES

CATS

ACEATOMIC

PHYSICS

DATA*

•TAPS

'LINES

•NERD

•USERS

PARADISE FILES

Central to the system is the data file. The

arrows indicate that the CATS code writes

data to the file, the ACE code reads and

writes data, and the other codes read various

"data. The LINES code generates synthetic

spectra and the NERD code is a non LTE

code; these two codes will not be discussed

further here.

The sets of commands can be sent to the

CATS, ACE, and TAPS codes either through

interactive use of the computer terminal or through a text file. Full descriptions of all commands are available in the manuals. ^''^'^ Running the three codes with these sets of

commands will ultimately produce DCS's and

EICP's for the 6s"-6s6p ^Pi transition in neutral barium. The CATS run takes ~ 30 sec.
24

ACE takes ~ 180 sec and TAPS takes ~ 1 sec of Cray-1 time for this test case. Scattering calculations were performed at 20 and

50 eV with 250 partial waves at each energy.

The setl command in ACE was used to select

the 6s--6s6p ^Pi transition. Other level to

level transitions may easily be selected. Tables 1-4 show samples of the EICP's for bariuim. Table 1 shows the EICP's from the

blum file for 50 eV. Table 3 shows the cor- responding fano file; Table 3 shows the hertel

file; Table 4 shows the stokes file. The transition energy for barium 6s'-6s6p ^Pi is 2.4

eV so that 50 eV electron energy corresponds

to over 20 times the threshold energy. This

requires a large number of partial waves for

convergence. The following figure shows the

orientation parameter for different numbers

of partial waves for this case.

Ba 6s^ - 6s6p 'P 50 eV

160

' l.,.-260 ^

I...-120 —

0 0 20 0 40 0 to o ao a looo uoo uoo ito o itoo Angle (deg)

Excitation to a specific magnetic sublevel can also be calculated from our codes. From

these, polarization fractions can be obtained.

Table IV is taken from Ref. 12. It shows colli- sion strengths to magnetic sublevels for neonlike molybdenum (Mo"'"'^^) calculated with

the fully relativistic code of Zhang et al.-^^ compared with the present results. The collision strengths and cross section are related

by

where Q is the cross section, a^, is the Bohr

radius, w, is the statistical weight of the ini- tial state, E(Rj,) is the impact electron energy

in rydbergs and is the collision strength.

In summary, we have developed a set of

computer codes for atomic physics calculations at Los Alamos. These codes can cal- culate a large variety of data with a minimum of effort on the part of the user. In particular, DCS's and EICP's can be readily

obtained for arbitrary ions or atoms. Currently, the theory consists of non-relativistic

Hartree-Fock structure calculations and non

relativistic DWA or FOMBT collisional cal- culations.

This work was performed under the aus- pices of the U.S. Department of Energy.

REFERENCES

1. J. Abdallah, Jr., R. E. H. Clark, and R. D.

Cowan, Los Alamos Manual LA-11436-M

Vol. I (1988).

2. R. D. Cowan, Theory of Atomic Structure

and Spectra (University of California, Press,

Berkeley and Los Angeles, 1981).

3. R. E. H. Clark, J. Abdallah, Jr., G.

Csanak, J. B. Mann, and R. D. Cowan,

Los Alamos Manual LA-11436-M Vol. II (1988).

4. J. B. Mann, At. Data Nuc. Data Tables,

29, 407 (1983).

5. R. E. H. Clark, J. Abdallah, Jr., and S. P. Kramer, Los Alamos Manual LA-11436-M

Vol. Ill (1988).

6. G. Csanak, H. S. Taylor, and R. Yaris,

Phys. Rev. A, 3, 1322 (1971).
25

7. G. Csanak, H. S. Taylor, and R. Yaris, 10.

Adv. At. Mol. Phys., 7, 287 (1971).

8. K. Blum, F. J. da Paixao, and G. Csanak,

J. Phys. B, 13, L257 (1980). 11.

9. U. Fano and J. Macek, Rev. Mod. Phys.,

45,553 (1973). 12.

H. W. Hermann and I. V. Hertel, Comm.

At. Mol. Phys., 12, 61 (1982); ibid 12, 127

(1982).

N. Andersen, J. W. Gallagher, and I. V.

Hertel, Phys. Repts. 165, 1 (1988).

H. L. Zhang, D. H. Sampson, and R. E. H.

Clark, Phys. Rev. A, (submitted 1989).
26

tabic 1. Blua fil« for barium at 50 aV.

start 5 S.OOOOtt-i-Ol

th«ta

0.0000«-t>00

3.0000«4-00

1.0000«+01

1.5000«+01

2.0000«+01

2.5000«-i>01

3.0000«-t-01

lambda

1.0000«-»>00

4.4039«"02

7.7861«-02

3.3075«-01

4.7757«-01

4.1853«-01

3.9709«-01

Chi

0.0000«>00

5.0521«-01

•1.4757«H'00

•6.8734«-01

-5.8229«-01

•5.6558«>01

cos delta

0.0000«<»-00

i.oooo«-«-oa

i.oooo«+oo

9.9999«-01

1.0000«-»>00

1.0000«'fOO

9.9998«-0I

cos cps

0.0000c<»>00

1.0000«-fOO

1.0000«-^00

9.9999«-01

9.9999«-01

l.OOOOc-fOO

9.9999«-01

Table 2. Fane file for barium at SO eV.

start 11,12, c th«ta o 0.0000«4>00 0.

5.0000«-t>00 9.

l.OOOOefOl 2.

l.SOOOc-t^Ol 4.

2.0000«^01 3.

2.S000«-»>01 2.

3.0000«-»>01 2.

«- 1 5 S.0000«4>01

1- al-t- aO

0000*^00 0.0000«>00 -1.0000«400

930€«->02 1.79SS«-01 4.3394«-01

6769tt-01 «1.1917«-02 3.832U-01

I83S«-01 4.4€72«-02 3.I7SS«>03

8731«-01 3.1S42«»01 »2.1$35<i-01

7129«-01 4.1202®"0i "1.2780«-0l

I221««01 4ci310««>@l »9.5i39«»-02

a2't- 0.0000«-i>00

-4.77§S«-01

-4.il67«-01

-3.34(2«-01

-2.«121«-0l

-2.9073«-01

-3.0145«-01

Table 3. Hertcl file for bariua at 50 eV.

start 11^ e theta

O.OOOOe-^00

S.OOOOe-t-OO

1.0000e4>01

1.5000e-»-01

2.0000e-t>01

2.5000e-*>01

3.Q000e-^01

12, e" 1

rhoOO

9.53€3e-10

4.5688e-08

7.0877e-07

2.SS47e»0€

1.724€e-0C

9.3473e-07

2«0884e-0«

S.OOOOe^Ol

gaana(rad) ga»a(deg) 1 perp 1 perp

-4.3672e-09 -2.5022e-03 0.0000e-(>00 O.OCOOe^OO

«1.3832e>00 -7.92S3e40]. 1.98(le-01 1.9861e-01

1.55e7e^0Q 8.9191e401 5.3538e-01 5.3538e-01

-1.4418e400 •>8.2S07e>01 9.3(71e-01 9.3671e-01

€.3243e-0l -8.2089«-01 •>4.7034e^01 7.74€2e-01 7.7461e-01

8.4000e-01 -8.8300«-01 -S.0592e^01 5.4259e-01 5.4259e-01

8.314Se-01 -9.0747e~01 -S.1994e401 5.2442e-01 5.2442e-01

1

9.8008e-01

8.44€le-01

3.5009e«01

Table 4. Stokes file for bariua at SO eV.

start ll,i2,e® 1 5 5.QO0O«4.O1 c theta pi p2 p3

0 .OOOOe-t-OO 1.,0000e-t>00 -8.7344«-Q5 0. OOOOe-t'OO

5 .OOOOe-t-OO -9..1192e-01 -3.5910«-0l -1. 98€le-01

1 OOOOe-fOl -8, 4428e<-01 2.3834e«0a -5. 3338e-01

1..SOOOe-t-Ol -3, 3850e-01 -8.9344e«02 »f. 3€71e-01

2..00G0et>01 -4. 48S0e-02 «€.3084e»01 -7. 74€2e-01

2. S000e-»>01 -1. 6294«-01 °8.2404e-01 -5. 42S9e->01

3 OOOOe-^01 -2. 0S8l««01 -8.2€21e-01 -5. 2442e-01
27

TA8L1 IV. CoaparlMii of rt«ulc« for cha coUialon •trtn«th« for •xctcatloa

froa ch« grouad co ch« a«(n«cic subl«v«i« Mf of ch« •zclctd odd p«rlC7

l«v«is wlch a 3 «ad J{ I la a«oa-Lik« aoiybdcauB. la each caa« upp«r cacrlas

ar« ch« pr«««at fullr r«l«clvlscic raaulta aod ch« sacoad ancrlaa ar« froa cha coda of aaf. 13. Nvaibara la squara brackaca daalgaaca povara of 10 by which

adjaeaac aacrlaa ahouid ba auidpliad.

Laval «f 10 30 200 500 1000

2p3a 0 3.43(-4] 4.63(-4j 8.991-41 1.48(-3! 2.00t-3j or

^ 3.47(-4j 4.a0(-4] 9. 20 (-41 1.46(-31 1.87[-31

(In 3a ) 1 l.llt-41 1. 28[-4l 2.301-41 4. 83 [-41

1.08C-*! 1.29 (-41 2.3«(-4l 3.03(-4l 9.13(-4l

2p3a ^P. 0 2.3«(-4l 3.06(-41 3.61(-41 9.17(-41 1.24(-31 or

^ 2.671-41 3.36[-4l 6.60(-4l 1.03(-31 1.36(-31

1 1.19(-4l 1.23 (-4] 1.6S(-41 3. 00 (-41 3.31[-4l

l.23t-4l 1.32 (-41 1.8»(-41 3. 36 (-4] 6.32(-4l

2P34 ^ 0 2.14(-4] 1.78 (-41 1.08(-41 6.90(-5l 5.23(-5l or

^ 2.0«(-4] 1.67(-41 9.ai(-s] 3.77(-5l 3. 81 (-31

^^'3/2-*'3/2'l 1 7. 94 (-4] 6.30(-4l 3. 13 (-41 1.2S(-41 5.72(-51

8.03(-4] 6.33(-4l 3.U(-4l 1.19 (-41 4.93(-5l

2p3d ^Oj^ 0 1.34 (-2] 1.61(-21 2,33(-2l 3.U(-21 3. 73 (-21

1.371-21 1.62 (-21 2.31(-21 2.99(-2l 3.43(-21

^2^3/2^5/2^

1 6.03(-31 6.61(-31 8.95(-3l 1.37(-21 2. 06 (-21

3.911-31 6. 30 (-3) 8.84(-3l 1.37(-21 2.06(-2]

2p3d H 0 1.13(-21 1.36 (-21 1.97(-21 2. 65 (-21 3. 22 (-21 or 1.18t-2l 1.39(-2l 1.99(-21 2.62(-2l 3. 03 (-21

^^'1/2^3/2^1

1 S.00(-3I 5.49(-3l 7.U(-31 1.1* (-21 1.73{-21

4.97t-3l 3.44(-3l 7.44(-31 1.16(-21 1.76(-21

2a3p ^P, 0 2. 30 (-41 3. 37 (-41 6.43(-41 1.08(-31 1.48(-31 or 2.221-41 3. 01 (-41 3.79(-4l 9.471-41 1.26(-31

2«3fS.

or ^^1/2^3/2^

1.36(-41

1.46 (-41 -

6.30(-4I

7.73(-41

1.93(-41

2.'92(-4j1.37(-41

lw*4r-4l

8. 78(-4l

1.03 (-31

2.33(-4l

2.80(-4]

1.93(-41

1.73(-4l

1.74 (-31

2.0O(-3l

4.17(-41

4.89(-4l

3. 31 (-41

3.00(-4l

2.97(-3l

3.23(-3l

8.39(-4l

9.97(-4l

3. 82 (-41

3. 33 (-41

4.13(-31

4.30(-3l

1.381-31

1.8a(-3l
28

Systematic Investigations of Mott Polarimetiy with Very Thin Gold Films . M.A.Khakoo, J.A. Brand, W.M.K.P.Wijayaratna and T.J.Gay . Physics Department, University of Missouri-RoUa, Rolla, MO 65401 .

1 INTRODUCTION.

The precision determination of the absolute spin polarisation of a collimated beam of

electrons by double scattering measurements has been the subject of considerable interest in the

past^. Recently the Munster groups have used an optical helium polarimeter^ to calibrate a Mott polarimeter absolutely to better than ± 1%, providing that the accuracy of the helium

polarimeter is only limited by the statistical precision in the determination of the circular

polarisation of the emitted light and the r? ^/F theoretical analysing power^. The question of the

absolute precision of the helium polarimeter still remains experimentally untested.

A complementary problem is to determine the ultimate precision of a retarding potential

Mott polarimeter, if possible, to better than 1%. This problem has been carefully considered by

Fletcher et al.^ , who experimentally demonstrated a limitation of ±5% absolute accuracy in their

Mott scattering experiments.This limitation was "a consequence of foil thickness dependence of

the Mott asymmetry (±3%) and the theoretical uncertainties in the calculations of the Sherman

function(±2%)"'».

One motivation for making a sub-1% relative Mott asymmetry measurement is to determine the absolute Sherman function for Mott scattering at a given incident electron energy

to a considerably higher precision than present theory allows, using the helium polarimeter as the

absolute standard. Of course, agreement between the two methods, using available theoretical

values of the Sherman function, will give us increased confidence that our assumption of absolute

accuracy for the helium technique is justified.

However, in order to improve the present state of Mott scattering measurements a clearer understanding of multiple and plural scattering is required to quantify the present

disagreements between theory and experiments. For example, a Monte-Carlo calculation'' of

elastic plural electron scattering in gold films showed that the asymmetry for films of, e.g., 2()()a at the incident electron energy of 3{)keV should be reduced by as much as 50% by plural elastic

scattering in the film, a fact which is not in agreement with existing Mott asymmetry

measurements^

.

It is to shed new light on this particular aspect of Mott scattering from gold films that this

work is aimed.

EXPERIMENTAL APPARATUS.

The Rolla polarised electron source consists of a continuously cesiated NEA-GaAs

photoemission source pumped by a GaAlAs solid-state laser ( 7960a). The laser was -95%
29

circularly polarised by using an achromatic A/4 plate. The source was found to be extremely

stable for periods in excess of several months, with an average electron spin polarisation of

-30%. The electrons were transversely polarised via a 90° electrostatic spherical bender,

collimated and transported into the Mott chamber by cylindrical electron lenses.Base pressures in the source and Mott chambers were typically 4x10"^ Morr and ZxlO'^torr ,respectively,rising to

Ix 10"^ °torr and IxlO'^torr during experimental runs. The Mott detector (Farago/Rice design*5)was UHV compatible and had two high

resolution retarding potential detectors at ± 120° scattering angles. The design of the retarding

optics included (among other collimating apertures) two apertures 2mm in diameter placed 6mm

apart where the retarding potential was applied, to produce a spatially well defined retarding

potential. The retardation optics performed with an energy resolution of ±I.5eV FWHM. The

angular resolution of the detection system was ±0.3° FWHM. After the retardation optics the

scattered electrons were accelerated to kinetic energies > 135eV at the entrance nose cone of the

channeltron detectors in an effort to minimise systematic variations of the detection efficiency as

a function of energy loss. Five gold films vacuum-evaporated onto Formvar were used. A bare

Formvar foil was included for background determination purposes. The gold film thicknesses

were 10,25,50,100 and 200a .The relative error in film thickness is estimated at ±8%.

The electron beam was dumped into a 150mm long, 25mm diameter Faraday cup and the

current was continuously monitored using an electrometer connected by an IEEE bus to a personal computer. The computer controlled a stepper motor which rotated the A/4 plate to reverse the electron spin polarisation as well as a ramp vdltage output (±2000 ± lOmV) for the

retarding potential detectors. The Mott scattering asymmetry is evaluated by taking the counts in

the left (L) and right (R) detectors for four orthogonal positions of the A/4 plate, corrected for

Formvar and X-ray backgrounds.The left-right asymmetry, A, is given by

The ± superscripts represent spin-up/down electrons , respectively. This method of evaluating

A eliminates instrumental asymmetries^. The asymmetry is related to the Sherman function, S, (provided no plural scattering processes are present) by

A = (1-C )/(!+£) (1)

where

e = J{(L + R-)/(R + L-)}. (2)

A = S-Pe (3)

where ?q is the transverse electron spin polarisation.
30

For all the measurements of signal + background a corresponding background run was made using the Formvar foil (corrected for the variation of the electron current and taking into

account the transmission of the foil in question). This was crucial to reduce systematic errors in the experiment since background scattering due to Formvar could be a significant fraction of

that due to scattering from the gold. At worst this was approximately 23% for a lOA film at 20kev

incident electron energy and 1300eV maximum energy loss (retarding voltage).

Using this system, Mott scattering asymmetries were measured at various incident

electron energies from 20 to 120 keV as a function of film thickness and maximum energy loss. Asymmetry vs. film thickness measurements were carried over periods not exceeding a day to normalise the different asymmetry values to a fixed electron spin polarisation. It was observed

that the electron spin polarisation did not vary by greater than 1 % over this period.

14 1

12 "..1 I

'

[

I—

<

iO

tr

8

ill

:e

6 K)0

AS 10 A

< 4

o 1 1 _i 1

1 1—

0 400 800 1200

ENERGY LOSS. AE (eV)

Figure 1. Moll asymmclry vs. maximum energy loss (or lOA Him thickness M an incident electron energy ot lOOkeV, showing the lack of dependence of asymmetry on energy loss.See text for details.

RESULTS.

Figure 1 shows the asymmetry plotted against maximum energy loss (retarding

voltage) for our lOA gold film at the incident electron energy of lOOkeV. From figure I it can be seen that the asymmetry vs. maximum energy loss is effectively flat with a nominal slope of

-l(±2.3)x 10'^ V'l. This implies the absence of multiple inelastic scattering i.e. the asymmetry

remains within ±0.1% for an asymmetry of O.i .More importantly, it implies that using a retarding

Mott polarimeter at lOOkeV to make foil thickness measurements of the asymmetry at non-zero
energy loss retardation values will yield the correct extrapolated asymmetry free from multiple

scattering. This is an advantage since the larger energy loss retardation values provide

correspondingly higher electron count rates due to improved transmission through the retarding

optics. We have used this fact to make relative asymmetry foil thickness measurements (following

the analysis of Fletcher et al.^) to an accuracy of ±0.7%, i.e. considerably better than that of

Fletcher et al (±3%). Details of this analysis are lengthy and will not be described here ,but will

be presented at the meeting.

With the retardation at 4eV energy loss (which essentially limits us to observing

elastically scattered electrons only) we have made relative measurements of Mott scattering

asymmetry vs. film thickness vs. incident electron energy.These data are shown in figure 2. The

^0 50 100 150 200

FILM THICKNESS, t(A)

Figure 2. Motl asymmelry vs. film thickness for various Ej values. Key:- O Ej = lOOkeV, E| = 6()keV,

A Ej = 40kcV, • Ej = 20kc V. The curves are fits lo the experimental points using equations 4 and 5 .

curves are our semi-empirical extrapolations derived from the observations of Hnizdo"'. For a given value of incident electron energy (Ej) his Monte-Carlo calculations showed that the film

thickness (t) dependence of the asymmetry A for elastically scattered electrons can be described

by

A(t) = A(t = 0)/(l + b(Ei)-t), (4)
32

where b is independent of foil thickness. However, since he did not account for inelastic

processes in attenuating elastically scattered electrons emerging deeper in the film, a modification of equation 4, replacing t with an effective film thickness t ' given by

t' = b(Ei). L- (l-exp(-t/L)) (5)

is required. L is the mean free path for inelastic scattering. The term l-exp(-t/L) is the reduction

of elastic electrons by inelastic scattering as a function of film thickness. The validity of equation

5 can be observed in figure 3, where the observed intensity of elastically scattered electrons are plotted against film thickness for several Ej values.

ID

>-tr 0 <

ir

CD

cr<

20keV

40keV

60keV

100 keV

'0 50 100 150 ZOO FILM THICKNESS, t (A)

Figure 3. Nj„, (count rale of scallered eleclrons) vs. film thickness for the maximum energy loss value of 4eV

(essentially only clastic electrons are detected) and various Ej values. See text lor details.
33

It is clear from figure 3 that, eg., at 20keV incident electron energy, the lOOA film is

infinitely thick for elastic scattering processes ,a fact that is also clearly borne out in the film

thickness dependence of the Mott scattering asymmetry which effectively bottoms out at this film

thickness. This is due to the fact that the inelastic mean free path is short enough (L-2()a at 2()keV) to stop deeply penetrating electrons from emerging elastically from the gold film at the

scattering angle of 120°

.

Normalising the lOOkeV extrapolated asymmetry to the Sherman function value of 0.394

of Ross and Fink'^ gives (using the film thickness extrapolated values at the other lower energies)

the values of S(20keV) =0.312±0.014, S(40keV) = 0.333±0.010, and S(60keV) = 0.353±().00cS, in good agreement with theory. The experiment of Campbell et al.^ which made erroneous asymmetry vs. energy loss extrapolations is the only other measurement which shows similar

agreement with theory. We have correctly modified their S values based strictly on the

assumptions of their paper and find that at 20keV ,for example, this corrected value falls 6.6%

lower than theory, showing indeed that their data did, like all other previous experiments, suffer

from plural scattering.

In conclusion, the above work strongly suggests that in the near future it will be possible

to make an absolute determination of the Sherman functions for Mott scattering from a solid

target, using the helium polarimeter as a calibration standard, to a sub-1% precision and hence

extend the precision of absolute Mott polarimetry to an accuracy better than that provided by

present theoretical values of these Sherman functions.

* Supported by the NSF (Grant PHY-8602066)

REFERENCES

( 1) See for example H.Boersch et al.(1971) Nucl.Phys. A163, 625

and H.Hopster and D.L.Abraham ( 1988) Rev.Sci.Inst. 59, 49

(2) M.Uhrig et al.( 1989) Rev.Sci.Inst. 60, 872

(3) T.J.Gay (1983) J.Phys.B.Atom.Mol.Phys. 16, L553

(4) G.D.Fletcher et al.(1986) Phys.Rev. A34, 9 1

1

(5) V.Hnizdo (1973) Nucl.Inst.Methods m503

(6) UA.Hodge et al.( 1979) Rev.Sci.Inst. 50, 5

(7) A.Ross and M. Fink (1988) Phys.Rev A^, 911

(8) D.M.Campbell et al.( 1985) J.Phys.E Sci.Inst. 18, 664
34

POLARIZATION CORRELATION EFFECTS IN THE TWO-PHOTON DECAY OF METASTABLE

ATOMIC DEUTERIUM

A J Duncan

Atomic Physics Laboratory, University of Stirling,

Stirling FK9 4LA, Scotland

Over the past few years a series of measurements"' .2,3 have been carried out to investigate the polarization correlation properties of the

two photons emitted in the decay of metastable atomic

deuterium. In the first of these experiments in which

two polarizers were placed diametrically one on either

side of the source, the measurements verified the

quantum mechanical predictions and violated Bell's inequality^ thus ruling out a local realistic interpretation

of the results. However, because of the low detection

efficiency of the photomultipliers only a small sample of the photon pairs emitted by the source was detected

leaving open the possibility that, in some way, a biased

sample of pairs was detected. The assumption that such a bias does not exist is known as the no- enhancement assumption. 5 if this assumption is not made Garuccio and Selleri,6 for example, have shown

that it is easy to explain the results of all existing two

polarizer experiments in local realistic terms by

attributing to a photon an additional (hidden) parameter

which determines its probability of detection. However,

their model leads to a direct conflict with the predictions

of quantum mechanics in experiments in which an

additional linear polarizer is inserted on one side of the

source. Such an experiment with an additional linear polarizer has now been completed3 and the results once again have been shown to be in good agreement

with quantum mechanics and to rule out local realistic interpretations of the kind suggested by Garuccio and

Selleri. In addition, in evaluating the quantum

mechanical prediction an interesting question is raised

regarding the coherence length to be attributed before

detection to single photons which are an integral part of a two-photon excitation.

In this experiment the polarization correlation

properties of the two-photon radiation from a metastable

atomic deuterium source are investigated using three

pile-of-plates polarizers as shown in Fig 1

With the transmission axis of polarizer a held fixed in the x direction, the transmission axis of polarizer b is rotated through an angle p in one sense and the

transmission axis of polarizer a' through an angle a' in the opposite sense. The ratio R(P,a')/R(P,«>) is then

measured as a function of the angle a' for various

SOURCE

hv-i

/

LINEAR POLARIZERS

hV2

y

a'

FIG 1. Diagram to illustrate the three-polarizer

experiment. The source consists of a beam of metastable atomic deuterium, the two-photon radiation

from which is detected and analyzed by a conventional

electronic coincidence technique.

angles p, where R(P,a') is the rate of coincident detection

of photon pairs with all the polarizer plates in place, while

R(P,«>) is the rate with the plates of polarizer a' removed. If the action of the polarizers a, a' and b on the state vectors

of the photons which are incident upon them is represented, respectively, by the 2x2 matrices A, A' and

B, then, according to quantum mechanics, the ratio R(P,a')/R(P,oo) is obtained from the 4x4 density matrix p

representing the two-photon state by the formula

R(P,a') ^ Trace [A'A<g Bp(A'A®B)t]

R(P,°°) Trace [A® Bp (A® B)t]

or alternatively

R(p,a') Tracei (A'ApeffAtA't)

(1)

R(PH Trace i (ApeffAt) (2)

In expression (2) the traces are taken only over the

polarization variables of photon 1 (on the right in Fig 1) and pgff, the effective density matrix obtained by taking

the trace over the polarization variables of photon 2 (on

the left in Fig 1), is given by

Peff = Trace2(BpBt). (3)

In a sense then, expression (2) demonstrates that, for a given two-photon state, the polarization properties of
35

photon 2 on the right can be considered to be

determined by the properties and setting of the

polarizer b on the left. If M/^ and are the

transmission efficiencies (the moduli squared of the

transmission amplitudes) for light polarized parallel to (the wanted component) and perpendicular to (the

unwanted component) the transmission axis of polarizer

a, respectively, with similar definitions for Mg, mg, M/^-

and m^', then from either expression (1) or (2)

R(p,a')/ R(PH = F + G(P) cos2a' - H(|3) sin2a" cos <]) (4)

where F = 1/2 (M^- + m/^-),

G(P) = 1/2 (Ma- - niAO [( MaP - mAQ )/( MaP + hiaQ)]

H(P) = (MAmA)''/2 [(^^ . mB)(MA-mA-) /

2(MAP-KTV\Q)]sin213

with P = 1/2 [(Mg+mg) + (Mg-mB) cos 2|3]

and Q = 1/2[(MB+mB) -(MB-mB)cos2p.]

The term H(P) sin 2a' cos(t> is of particular interest and results from the interference between the wanted

light and the unwanted light passing through polarizer a. The angle (}> represents the relative phase between the

complex transmission amplitudes for wanted and

unwanted light through polarizer a, and, for light passing

directly (ie without internal reflections)through the

polarizer, it would be expected that (t)=0°, cos 0=1.

However, in practice, the effect of internal reflections

between the polarizer plates cannot be ignored. It is assumed that the contribution of internal reflections to the wanted component is negligibly small since they occur near to Brewster's angle, but a significant part of the unwanted component does arise from these

reflections. However, because of the small random

deviations of the plate alignment from Brewster's angle,

the lack of parallelism of the surfaces of the individual

plates and the imperfect polish of the plate surfaces

(only 2X at 243nm) it is unlikely that the component of the unwanted light resulting from internal reflections will interfere with the light passing straight through the

polarizers irrespective of the coherence length to be

associated with the photons passing through each

polarizer. It can be argued that this coherence length is extremely short since the source has a large bandwidth

(121.5nm - 0°) with wavelengths in the range 185nm to 355nm being detected in practice. Taking the

coherence length Ic = 'k^/AX where X is the mean

wavelength and AX the bandwidth gives Ic.r350nm,

which is of the order of the wavelength of the radiation

itself. With such a short coherence length, interference

of the internally reflected light with the straight through

component would thus not take place. If such

interference does not in fact occur then the factor

(MAmA)''/2 in the expression for H(P) must be modified to (MAhA)"'/2 where hA represents the transmission

efficiency for the unwanted component not resulting from

internal relfections. The quantity hA is wavelength

dependent and may be calculated from a knowledge of the optical properties of the polarizer plates and the

spectral distribution of the source to have the value

0.0182. In a subsidiary experiment described

elsewhere^ the following values were found for the

transmission efficiencies Ma= M b

= 0.908±0.01 3,

Ma = Mb= 0.0299± 0.0020, Ma' = 0.938±0.010,

mA'=0. 040+0. 002. In practice, however, since the ntio

R(P,a')/R(p,c«) is wavelength dependent, and depends

nonlinearly on wavelength dependent transmission

efficiencies a wavelength averaged value was computed.

The result of these calculations is shown as the quantum

mechanical prediction in Figs 2 and 3 but the curves do

not, in fact, differ significantly from those obtained by

direct substitution in expression (4) of the measured

transmission efficiencies quoted above.

01 1 I I I 0 225° 45° 675 90° ANGLE OC'

FIG 2. Variation of the ratio R(p,a')/R(P,«>) as a function

of a' for p=0O (O), 33° (•), and 67.5° (A). The solid curves represent the quantum-mechanical predictions.
36

The experimental results for (3=0°, 33° and 67.5°

are shown in Fig 2 and, clearly, within the limits of experimental error, agree well with the

theoretical. predictions.

Also, referring to equation (4), it is clear that the ratio R(p,a)/R(P,°°) should not be symmetrical with respect to a change in the sign of angle a' because of the

presence of interference between the wanted and

unwanted components of radiation transmitted through

polarizer a. To investigate this prediction an

additional measurement was made for p = 67.5°, a' = -45° and the result is shown, in comparison to the result for positive a", in Fig 3 along with the corresponding

quantum mechanical prediction.

FIG 3. The ratio R(p,a')/R(p,c«) as a function of a' for P=67.5°. The upper (lower) curve is the theoretical

result for a' <0 (a'>0). The experimental points are marked o for a'<0, A for a'>0.

These results are also of interest because of their relevance to the suggestion by Garuccio and Selleri,^

based on a particular class of local realistic models, that a

finite difference might exist, in this situation, between

the predictions of quantum mechanics and local realistic theories involving enhanced photon detection made

possible by the low detection efficiency of photomultipliers. In this class of theories, capable of explaining all existing two-polarizer experiments,

Garuccio and Selleri postulate that a detection vector X, in addition to a polarization vector I is to be attributed to each photon of the pair emitted by a two-photon

source. The detection vector X is assumed to be

unaffected by passage through a linear polarizer and the

probability of detection of a photon is assumed to depend

on the angle between i and X. Using this approach, for the parameters of the present experiment the theory of Garrucio and Seilen set an upper limit of 0.413 on the ratio R(33°, 38°)/R(33°,oo) whereas the actual experimental

point has the value 0.585±0.029 violating their prediction

by over six standard deviations. More recently, Selleri^ taking other factors into account set an upper limit of 0.514 on the ratio which is still, however violated by the

expenmenlal result by almost three standard deviations.

The three polarizer experiment, therefore, provides

strong evidence against the possibility of enhancement in the detection process depending on the existence of an

additional (hidden) parameter, and appears to rule out the

class of local realistic theories proposed by Garuccio and

Seileri.

Experiments have also been carried out to investigate

the effect of a weak electric field on the polarization

correlation properties of the two photon radiation.

Unfortunately, although the theory of the two-photon

decay of metastable atomic hydrogen was developed some time ago,^'^ and the effect of an electric field on

the lifetime of metastable atomic hydrogen is well

known, "1 0 no theoretical treatment exists to describe the

effect of an electric or magnetic field on the polarization

correlation properties of the two-photon radiation emitted

by metastable atomic hydrogen or deuterium. It is perhaps interesting to note, however, that some

theoretical and experimental work has been carried out

with reference to the polarization correlation properties of the two photons emitted in cascade by excited calcium

atoms in a magnetic field. ^

In the case of atomic hydrogen or deuterium the

matrix element describing the two-photon decay in the

absence of fields is in the form

<¥f|r.ei|\i/j><vjlr.e2!vt/i> ^<yf|r.e2|\j/j><>Fjii eil¥i: Wi-Wf-hu2 Wj-W}-hi>i

where |\|/j> and |\|/f> represent the initial (2S) and final (1S)

state of the atom, Wj and Wf the initial and final energies

and Wj the energy of the virtual intermediate states which

are the P states of the atom. The unit vectors e-| and 62

are in the directions of polarization of the two photons of frequencies di and vp- At first sight then, on

examination of the matrix element it might seem unlikely

that a weak electric and/or magnetic field would have any

significant effect on the two-photon decay process since

Wj and Wf will only change by a small amount in
37

However, it is possible tiiat angular momentum effects could be very sensitive to the presence of a field since it will remove the degeneracy of the virtual intermediate

states j\|/j> and may result in a substantial change in the

polarization state of the two photons. The

characterization of the polarization state of the two

photons, of course, requires in general the specification

of 15 independent elements of a 4 x 4 Hermitian matrix,

although, in practice, this number will normally be

reduced by various symmetry requirements. Also, in the usual arrangement'' with polarizers placed

diametrically on either side of the source, say on the left and on the right, as has already been demonstrated in equations (2) and (3) the polarization properties of the

radiation on the right detected in coincidence with that on the left can be expressed3 in terms of an effective 2

X 2 density matrix whose three independent elements

depend on the two-photon polarization state and the

setting and properties of the polarizer on the left. Measurements of the effective density matrix for one

type and setting of the polarizer on the left then gives

partial information on the two-photon polarization state. The complete specification can be obtained by a series

of measurements of this kind with different types and

settings of polarizers on the left.

In a preliminary experiment,'' 2 referring to Fig 1, polarizer a' was removed. Polarizer b was fixed with its transmission axis in the x direction while polarizer a was

rotated through an angle a relative to the x axis in the same sense as for angle a' in the three-polarizer

experiment. An electric field of 30 Vcm"'^ was applied

to the source in the x direction by means of electric field plates of the Spiess^S design with transverse

dimensions 10 mm x 22 mm placed a distance 13 mm

apart. The measured variation with angle a of the ratio R(a)/R(oo), where R(a) is the coincidence rate with the

transmission axis of polarizer a at angle a and R(oo) is the

rate with the plates of both polarizers removed, is shown

in Fig 4, along with the experimental results and

theoretical predictions in the absence of the electric

field.

An additional measurement for a = -450 gave

R(-450)/R(oo) = 0.87±0.30. Measurements of right hand

and left hand circular polarization were also carried out by placing a quarter-wave plate in front of polarizer a. As

a result, the following values for the three effective Stokes' parameters were obtained.

= [1(450) - l{-450)]/ [1(450) + i(-450)j = - 0.248±0.170

P2 = [l(RHC) - l(LHC)]/ [l(RHC) + l(LHC)] = 0.014+0.159

P3 = [1(00) - 1(900)]/ [1(00) + 1(900)] = 0.93210.366

ANGLE cx

FIG 4. Variation of the ratio R(a)/R(oo) as a function of a

for zero electric field (O) and for an electric field of 30

Vcm'"' (9). The solid curve represents the theoretical

prediction for zero electric field.

The preliminary nature of these results must be

emphasized. For a definite conclusion to be drawn the

measurements need to be repeated and extended.

However, if they are confirmed these obsen/ations would

represent an essentially new effect in atomic physics.

Acknowledgements

The author wishes to acknowledge the collaboration

with his colleagues, H J Beyer, T Haji-Hassan,

H Kleinpoppen, E Merzbacher and W Perrie. In particular, he wishes to thank T Haji-Hassan for his agreeing to the presentation of unpublished results from

his thesis.
38

References

1. W Perrie, A J Duncan, H J Beyer and

H Kleinpoppen, Phys.Rev.Lett., 54, 1790 (1985).

2. T Haji-Hassan, A J Duncan, W Perrie, H J Beyer

(1987)

3. T Haji-Hassan, A J Duncan, W Perrie,

H Kleinpoppen and E Merzbacher,

Phys.Rev.Lett., 237 (1989).

4. J S Bell, Physics (N.Y.) 1, 195 (1964).

5. J F Clauser and M A Home, Phys.Rev.D., IQ, 526

(1974).

6. A Garuccio and F Selleri, Phys.Lett.A, 103 . 99

(1984).

7. F Selleri, private communication.

8. M Goppert-Mayer, Ann.Phys.(N.Y.) 2, 273 (1931).

9. G Breit and E Teller, Astrophys.J. Si, 215 (1940).

10. G W F Drake and R B Grimley, Phys.Rev AS, 157

(1973).

11. A Aspect, J Daiibard, P Grangier and G Roger,

Opt.Commun. 42, 429 (1984).

12. T Haji-Hassan, PhD Thesis, University of Stirling,

1988 (unpublished).

1 3. G Spiess, A Valance and P Pradel, Phys.Rev.A.

746 (1972).
39

ELECTRON IMPACT EXCITATION OF THE n = 3 STATES OF HELIUM

AND 5s3P^ STATE OF KRYPTON

Albert Crowe

Department of Pure and Applied Physics

The Queen's University of Belfast

Northern Ireland, UK

Since the original angular correlation

measurements of Eminyan et aU, excitation of

the 2iP state of helium has been studied

extensively using this technique and more

recently the polarisation correlation technique.

There has been a correspondingly large number

of theoretical studies of this state. Although

there are still major quantitative discrepancies

between theory and experiment over wide

ranges of both energy and angle for the 2iP

state, the emphasis of studies of this type in

helium has recently switched to the n = 3

states. Experimental studies have been

reported from Stirling, Utrecht, Perth and

Belfast while theoreticaal work on these states has been reported using the first—order many

body theory (FOMBT), distorted wave Born

approximation (DWBA) and the R—matrix

approach.

The four n = 3 states, iP,3P,iD,3D, have

now been studied experimentally over limited

ranges of energy and angle. The studies

date back to 1976 with the angular correlation

studies of Eminyan et al2 and the polarisation

correlation studies of Standage and

Kleinpoppen3. Studies of the other three states are more recent. The data have been analysed using the (A,

x) parameterization of Eminyan et al^ or

alternatively in terms of the charge cloud and

angular momentum description 4 (Pi in, Lx, 7, Poo)- The experimental techniques involve

measurement of either the angular correlation

between the scattered electron and the photon

arising from the decay of the excited state or the polarisation of the radiation in coincidence

with the scattered electron. Both the angular

correlation technique and a measurement of the

linear polarisation (the Stokes parameters Pi,

P2) of the radiation emitted perpendicular to the scattering plane map out the shape of the

charge cloud in the scattering plane. The

angular momentum transferred to the atom

during the excitation process can only be

obtained from a measurement of the circular

polarisation, P3, of the radiation emitted in

that direction. Additionally, for the D states,

there exists the possibility of exciting a dipole

perpendicular to the scattering plane and so complete information on the excitation can

only be determined by a linear polarisation

measurement, P4, in the scattering plane. For

the triplet states, allowance for the fine

structure depolarisation of the radiation must

be taken into account in determination of the

above parameters from the data.

Exdtation of the S^P state

To avoid complications, this state is best

studied by observation of the 501.6 nm

31P—21S) radiation rather than the 53.7 nm

31P-IIS) photons 5. The Stirling group2 3 6 7 have made measurements of both linear and

circular polarisation at energies of 50, 80, 120

and 160 eV for angles in the range 27.5° to 105°

while Beijers et al^ have measured the circular

polarisation at 50, 60 and 80 eV. Recent

angular and polarisation correlation

measurements in this laboratory ^.^ have

concentrated on the lower energy range, 26.5

and 29.6 eV, where comparison can be made

with R—matrix calculations i*^. While these

calculations are in qualitative agreement with

the experimental results for the Stokes

parameters Pi, P2 and P3 and hence with the

linear polarisation Pun and angular momentum

transfer L±, there is a curious discrepancy with

the data for the charge cloud alignment angle

7. 7 values are determined directly from the

minimum positions of the angular correlations

and are related to the measured finear

polarisations Pi and P2 through the

relationship

7-Itan-l(P2/Pi)

Figure 1 shows the experimental and

theoretical values of 7 as a function of electron

scattering angle at 29.6 eV and 26.5 eV. To

highlight the different variations, the

experimental data are plotted on a scale 0 to —1800 while the theoretical values are shown

over the range —90 < 7 < +900. Theoretically

there is an initial rotation of the charge cloud

to negative angles, in quafitative agreement

with experiment, followed by a rotation to
40

positive angles. Experimentally the charge

cloud continues to rotate more towards

negative angles before reaching a minimum.

1 1 1 1 26.5 eV1 1 1 1

0

/

1 J 1 1

/ ^ ^ o /

" -80

O

\ -120

-160

1 [ L 1 till

although the 19—state calculation gives a much

lower positive value than the 5—state for the

23P state. Theory predicts a double hump

structure for P3 which is also seen in the

experiment although displaced in angle and

much less pronounced.

0.5

-0.5

80

0

<u

>-

-80

-120

-160

-1 1 1 1

f n 29.6 eV

0.5

_1 ] I I L- 0 W 80 120 160 Elecfron scattering angle (deg)

Fig. 1. O Angular Correlation data^ • Polarisation Correlation data^

19-state R matrix 10

Excitation of the S^P state

The situation up to the last meeting in this

series was discussed by Beyer ii. Since then we

have measured 12 both angular and polarisation

correlations for the S^P state at 29.6 eV. Very

recently these have been complemented with

19—state R—matrix calculations.

Figure 2 shows the measured values of Pi,

P2, P3 obtained using both experimental

techniques compared with the 5—state R-matrix calculation for the 23P state^^ and

the 19—state R—matrix calculation 10 for the S^P

state at 29.6 eV. The negative values of Pi at 40 and 60° are not predicted theoretically,

0.0

-0. 1

/ /

0 20 40 50 80 100 120 140 160 1£ Electron scattering angle (deg)

Fig 2. O Angular Correlation data^^

9 Polarisation Correlation data

5-state R matrix 13 (23p)

19-«tate R matrix 10 (33p)
41

Excitation of the S^D state

At the last meeting considerable interest was focused on new results which indicated

that the sign of the circular polarisation P3 was

positive for small scattering angles, in contrast

to that measured for the P states. This was in

qualitative agreement with the FOMBT of

Cartwright and Csanaki^ in disagreement

with the 10 channel eikonal approximation

results of Mansky and Flanneryi'' and the

excited state potential DWBA results of

Bartschat and Madisoni^. Although

preliminary data from this laboratory's

confirmed the sign of P 3 as measured by Beijers

et al, both groups, after careful evaluation of

the A/4 plates used to determine the circular

polarisation, agree that the sign is in fact

negative2o, as for the P states, for small

scattering angles and energies around 40 eV.

Figure 3 shows the measured values of the

circular polarisation at an incident energy of 40

eV compared with the DWBA and FOMBT

calculations. It can be seen that although there

is qualitative agreement between the excited

state DWBA theory and experiment, there is a lack of quantitative agreement.

(.0 80 120

Elecfron scaHerinq ongle (deq)

Fig 3. # Present data

F0MBT16

DWBA-EPlS

DWBA-GPlS

Measurements of the linear polarisations Pi,

P2 and P4 and the circular polarisation P3 are presently being carried out for the 3iD state at both 29.6 eV and 40 eV, enabling all four

parameters describing the charge cloud to be

determined. The height parameter poo, calculated from the linear polarisations Pi and

P4 using the relation

3(1 +Pi)(l-P4)

2(4-(l-Pi)(l-P4)j

is shown in Figure 4 at an incident electron

energy of 29.6 eV. At all angles the measured

values of poo are small. The 10—state eikonal

approximation of Mansky and Flanneryi'' gives

values just greater than 0.2 in the range 40

<^<800. The deconvoluted n=3 electron

—cascade photon angular correlation data of van Linden van den Heuvell et al 27 indicates a value of 0.4 at 35 0 at this energy.

40 80 120 150

Electron Scattering Ang (deg) Fig 4.

Present data @ 29.6 eV

Excitation of the S^D state

Like the S^P state, both angular and

polarisation correlation measurements on the

33D state are made difficult by the large fine

structure depolarisation of the emitted

radiation. The S^D state depolarisation factors are 71/150 and 43/54 for the linear and circular

polarisations respectively.

Measurements of the Stokes parameters Pi,

P2, P3 have recently been reported2i from this

laboratory at 40 eV. The results are shown in

Figure 5 and compared with the only available

calculation at this energy and the preliminary

experimental data of Beyer 11. The effect of the

large depolarisation on the measured linear

polarisations Pi and P2 is obvious from their

small values. The Pi value of Beyer agrees well

with the present measurements while the

DWBA calculations disagree strongly with

experiment at large scattering angles. It is a general feature of all the correlation studies for

the n=3 states that the greatest discrepancies

between theory and experiment exist for P^

The DWBA calculations predict very small

values of P2 compared with experiment, while

the opposite is true for P3.

Poo
42

^2 0

-0.5

-0.5

-1.0

1 1 1

- \

r

1 1

1 1

1 1 1 J 1 1 1 1

1 1 1

} \ \ \ \ V

1 1 1

1 1

1 i 1 I 1 40 80 120 160 Eiecfron scattering angle Ideg)

Fig 5. % Present data^l

O Beyerll

DWBA-EPlS

DWBA-GPlS

Excitation of the 4p5 (2P3/2) Ss^Pj state of

krypton

We have measured double angular

correlations for the 4p5 (2P3/2) 5s3Pi state of

krypton at 15, 30 and 60 eV over a wide range

of electron scattering angles.

Two significant results are obtained.

(1) Excellent agreement is obtained

with the DWBA results of Bartschat and

Madison22, particularly for the most directly

measurable parameters. Figure 6 shows, as an

example, the situation for the alignment angle

7 corresponding to the position of the angular

correlation minimum at an energy of 30 eV.

This seems a surprising result when contrasted

with the situation in helium, but may be a

result of the dominance of the central potential

in the krypton case.

"^"^0

10 2D 30 AO 50 60 70 80 90 100 ELECTRON SCATTERING ANGLE (")

Fig 6. # Present results DWBA22

FOMBT23

(2) The rapid variation of the A

parameter (Figure 7) in the region of a

differential cross section minimum (Figure 8) at 60 eV. Experiments are now in progress to examine this phenomena at other electron

energies.

1-0

Tti 5t5 53 75 go gtr ELECTRON SCArTERING ANGLE W

Fig 7. ® Present results O King et al24 n Danjo et al25 O McGregor et al26 - DWBA22 FOMBT 23
43

electron scattering angle (deg.)

Figure 8.

Acknowledgments

This work has been supported by the

United Kingdom Science and Engineering

Research Council. The contribution of B.P.

Donnelly, S.F. Gough, P.B. Murray, and P.A.

Neiil to the work in this laboratory is gratefully

acknowledged.

References

1 M. Eminyan, K.B. McAdam, J. Slevin and

H. Kleinpoppen, Phys Rev Lett 31, 576 (1973).

2 M. Eminyan, K.B. McAdam, J. Slevin,

M.C. Standage and H. Kleinpoppen, J Phys B

8, 2058 (1975).

3 M.C. Standage and H. Kleinpoppen, Phys

Rev Lett 36, 577 (1976).

4 N. Andersen, J.W. Gallagher and

LV. Hertel, Phys Rep 165, 1 (1988).

5 P.A. Neill and A. Crowe, J Phys B 21,

1879 (1988).

6 K.S. Ibraheim, H.J. Beyer and

H. Kleinpoppen, Proc 2nd ECAMP,

Amsterdam, ed. A.E. de Vries and M.J. van

der Wiel, p 298 (1985); Proc 14th ICPEAC,

Palo Alto, ed. M.J. Coggiola et al, p 118

(1985).

7 H.J. Beyer, K. Blum, H.A. Silim,

M.C. Standage and H. Kleinpoppen, J Phys B

21, 2953 (1988).

8 J. P.M. Beijers, J. P. van den Brink, J. van Eck and H.G.M. Heideman, J Phys B 19, L581

(1986) .

9 P.A. Neill, B.P. Donnelly and A. Crowe, J

Phys B 22, 1417 (1989).

10 W.C. Fon, private communication.

11 H.J. Beyer, Correlation and Polarisation in

Electronic and Atomic Collisions, ed. A. Crowe

and M.R.H. Rudge, p 324 (1988).

12 B.P. Donnelly, P.A. Neill and A. Crowe, J

Phys B 21, L321 (1988).

13 W.C. Fon, K.A. Berrington, P.G. Burke

and A.E. Kingston, J Phys B 12, 1861 (1979).

14 G. Csanak and D.C. Cartwright,

Correlation and Polarisation in Electronic and

Atomic Collisions, ed. A. Crowe and

M.R.H. Rudge, p 31 (1988).

15 J. P.M. Beijers, S.J. Doornenbal, J. van Eck and H.G.M. Heideman, J Phys B 20, 6617

(1987) .

16 D.C. Cartwright and G. Csanak, J Phys B

2020, L583 (1987).

17 E.J. Mansky and M.R. Flannery, J Phys B

20, L235 (1987).

18 K. Bartschat and D.H. Madison, J Phys B

21, 153 (1988).

19 B.P. Donnelly and A. Crowe, J Phys B 21,

L637 (1988).

20 H. Batelaan, J. van Eck and

H.G.M. Heideman, J Phys B 21, L741 (1988).

21 B.P. Donnelly and A. Crowe, J. Phys B 21,

L697 (1988).

22 K. Bartschat and D.H. Madison, J Phys B

20, 5839 (1987).

23 G.D. Meneses, F.J. da Paixao and N.T.

Fadial, Phys Rev A 32, 156 (1985).

24 S.J. King, P.A. Neill and A. Crowe, J Phys

B 18, L589 (1985).
44

25 A. Danjo, T.Koike, K.Kani, H. Sugahara,

A. Takahashi and H. Nishimura, J Phys B 18,

L595 (1985).

26 I. McGregor, D. Hils, R. Hippler, N.A.

Malik, J.F. Williams, A.A.Zaidi and H.

Kleinpoppen, J Phys B 15, L411 (1982).

27 H.B. van Linden van den Heuvell, E.M.

Gasteren, J. van Eck and H.G.M. Heideman, J

Phys B 16, 1619 (1983).
45

Orientation and Alignment in Electron

Collisions with Sodium

P.J.O. Teubner, R.E. Scholten and G.F. Shen

Institute for Atomic Studies,

School of Physical Sciences,

The FHnders University of South Australia,

Bedford Park, South Australia, 5042, Australia

The electron impact of the 3^P state in sodium provides a particularly fertile ground for the

study of electron scattering from atoms. Whereas the analytical wave functions of the atomic states are not known as they are for atomic hydrogen, the very strong coupling between the ground and

the first excited state suggests that only a few states will be required in a close coupling solution of

the scattering problem. This description which was first alluded to by Seaton^, was subsequently

refined by Moores and Norcross^ and most recently has been applied in depth by McCarthy and

coworkers^.

When the Flinders coupled channels theory is tested against the experimental differential cross section^, the results are perhaps inconclusive. For example at an incident energy of 22.1eV, there

is excellent agreement out to 15° but the theory overestimates the cross section at larger angles. As

the energy increases the agreement between theory and experiment deteriorates. This observation

has led Mitroy et al.^ to question the validity of the experiments. We note that the theoretical

predictions are essentially invariant with respect to the inclusion of more states in the model and

by the use of an optical modeP.

Teubner et al} went to considerable effort to identify and remove possible systematic errors in

their experiments. The data were taken in the form of angular distributions and were normalised

to the total differential cross sections of Enemark and Gallagher^. The distributions are very

strongly peaked in the forward direction. For example at 22.1eV, 75% of the integral cross section comes from the angular range 0° < ^ 10°. Thus this technique requires measurements at very

small scattering angles with the consequent additional experimental problems of possible detector

saturation and interference by the incident electron beam. These problems were identified by

Teubner et al^ and were solved. The success can be inferred by the excellent agreement between

theory and experiment. What causes the lack of accord at larger angles? Are there systematic

errors in the experiments?

It is appropriate to ask for another test of the theory. This in part prompted the electron

photon coincidence experiments of the Fhnders group at several energies. In these experiments,

electrons which had excited the state and the photons arising from the decay of the state were detected in coincidence after the polarisation states of the photons were analysed. The

photons were detected normal to the scattering plane and a complete Stokes parameter analysis was performed^. The data from these experiments can be used not only to observe the shape,

alignment and orientation of the excited atomic state but also, through the concept of the reduced

total polarisation |P|, the role of spin in the collision can be studied.

Figure 1 shows the values of Pi, measured by electron photon coincidence techniques, compared

with the predictions of the 4 state close coupling theory of Mitroy et al? and with the superelastic

scattering data of Hertel and coworkers^. Although the coincidence measurements support the
46

theory out to 15°, the quality of the experimental data at the largest angle studied makes any

further comparison difficult. In any event it has already been observed that the kinematic region

of interest lies beyond 15° at 22.1eV. There are a priori reasons why the kinematic range of

coincidence experiments is limited. It can be shown^ that the counting time required to obtain

data of a specified relative error is proportional to the ratio ^ where Q is the total cross section

and a the differential cross section. At 22.1eV, (t(30°) is 130 times less than (j(10°). Thus it is unrealistic to use this technique at larger angles where the theoretical interest is greatest.

Figure 1. A comparison between Pi fronn coincidence experiments with theoretical predictions at forward

angles at 22.1eV.

The optical pumping technique developed by Hertel and coworkers^ appears to offer a solution

to this experimental problem. Time reversal invariance ensures that the data derived from optical

pumping experiments are equivalent to those which can be obtained from coincidence experiments.

That is e{Eo) + Na (3^5)

^ e{Eo - 2.1 eV) + Na (3'P)

Na (325)

Whilst the data shown in figure 1 do not appear to support this premise, there is no doubt

about its validity. In these experiments, radiation from a laser is used to pump a beam of ground state sodium

atoms from a specific HFS level [F = 2) to the F — 3 level of the 3^P3 state. An electron beam

of energy 20eV is introduced into the interaction region and an electron spectrometer views the

region of overlap of the electron, sodium and laser beams. The laser beam is introduced normal to the scattering plane. Electrons which have gained 2.1eV of energy in the collision are analysed by

the electron spectrometer and detected in a channel electron multiplier. The superelastic count

rate is then measured as a function of the polarisation state of the laser beam.

The equivalent Stokes parameters

^

7(0°) - 7(90°)

' 7(0°) + 7(90°) ^

7(45°) - 7(135°)

^ 7(45°) + 7(135°) ^

'
47

I{RHC) - IjLHC)

^ ~ I{RHC) + I{LHC)

can be measured, where I{a) is the count rate when the laser beam is polarised at an angle a

with respect to the outgoing electron beam direction.

Such measurements depend on the orientation and alignment of the sodium atoms induced by

the laser beam. These effects can be measured by appropriate observations^ of the fluorescence

radiation from the sodium beam and the appropriate corrections made. The resulting reduced

polarisations P,- are related to the measured values P, by

A.2

where the laser induced alignment and orientation parameters of the atomic beam are Oq'^ and

Oq^ respectively.

It is possible to compare the measured values P, with the theory. Figure 2 shows a comparison

of the linear component Pi with the predictions of the theory of Mitroy et al.^ . It is clear that

there is excellent agreement with theory out to about 20°. At larger angles the agreement is poor. A similar effect is observed with the other linear component P2. This lack of agreement is consistent with the description provided by the differential cross section.

ph 60

3 a: Pi,

2 + 15 o

ph P.

(2)

• Present results - 4CC Mitroy et a I

Figure 2. Pi as a function of scattering angle from optical pumping experiments at 20eV.

A clearer picture of the dynamics^° of the scattering process can be gained from the alignment

angle 7, the total linear polarisation Pi and the angular momentum L± transferred normal to the

scattering plane. These parameters are related to those in equation (2) by^°

= iarg(Pi + ^P2)

and L± = —P3
48

The angle 7 measures the angle between the major axis of the charge cloud and the incident

electron beam direction. The shape of the charge cloud is measured by in that the relative

length and width are given by ^-^y^ and ^—f^ respectively.

Measurements of the angle 7 are shown in figure 3 where they are compared with those pre- dicted by the coupled channels theory of Mitroy et al.^^ the distorted wave polarised orbital theory

of Kennedy et al}^ and with the First Born approximation. Also shown are the results of the

coincidence experiments of Riley et al.^ and the previous superelastic results of Hermann et al7

0 « Qq 90 135

Figure .3. 7 as a function of electron scattering angle compared with theory at 20eV.

It is immediately apparent that the results derived from the superelastic experiments are far

superior to that from the coincidence experiments. The kinematic range covered by the present

measurements covers the whole region of interest which is dictated by the theory. It is also clear

that the DWPO theory of Kennedy et al}^ is inadequate at all but the most forward of scattering

angles. This observation differs from the conclusion of Riley et al.^ and reinforces the well known

problem of forming conclusions based on limited data. The problems with the Born approximation

are also obvious. They appear to agree with the coincidence measurements for Be < 15° but this

is a measure of the quality of the data rather than an indication of any deep physical significance.

The agreement between the coupled channels theory and experiment is better for 7 than for either of the components Pi and P2. Although the prediction by the theory of the change of

quadrant is spectacular, there are significant differences in the range 20° < 6g < 50° and around

the minimum in the differential cross section. Whilst it is undeniably true that 7 gives a clearer

physical picture of the collision, it is a ratio measurement and the individual components Pi are a more sensitive test of the theory.

Figure 4 shows a comparison of the results of the measurement of Pg with those predicted by

the theory. The theory appears generally to be supported out to a scattering angle Og ~ 45° but

there are serious discrepancies for angles greater than this. The angular momentum transferred normal to the scattering plane is shown in figure 5. Again

there is a significant diiference between experiment and the coupled channels theory for scattering

angles greater than about 45°.

The data shown in figures 2-5 demonstrate that the problems which the coupled channels

theory had in describing the backward angle behaviour of the differential cross section are still present when a more sensitive test of the theory are applied.
49

Present results 4CC Mitroy et al

Figure 4. Pi as a. function of electron scattering angle compares with theory^ at 20eV.

• Present results

D Riley et a! 4CC Mitroy et a!

Figure 5. P3 as a function of electron scattering angle compared with a coupled channels theory^.

One can speculate about possible shortcomings of the theory. For example if exchange was not

adequately described, this influence would be more apparent at backward angles. This hypothesis can be probed by comparing the total reduced polarisation |P| with that predicted by the theory.

2 2 1^1 = [Pi + P2 + Pi

and |P| = 1 is a test of the coherence of the collision. It can be shown that, inter alia, the

coherent excitation of a doublet state impHes that exchange scattering can be ignored.
50

Present results 4CC Mitroy et al

Figure 6. A compaxison between measured values of \P\ and theory^.

Figure 6 shows the results of the present mecisurement of |P| compared with the predictions

of the coupled channels theory. Overall the agreement is excellent including the onset of the loss of coherence at about 60°. This corresponds to the minimum in the differential cross section and

the fact that the theory does not reproduce the point at 60° may simple reflect a slight shift in

the predicted minimum of the diiferential cross section. Consequently it would appear that the

difference between theory and experiment cannot be explained on the basis of exchange.

The variety of experimental evidence presented in the current study clearly demonstrates that

there are significant problems in the coupled channels theory. Tempting though it may be to retain the concept of few state coupled channel calculation to describe the excitation of the 3^P

state in sodium, the present results show that this model does not provide a complete picture of

the collision process.

References

1. M. Seaton, Proc. Phys. See. Lend. A 68, 457 (1955).

2. D.L. Moores and D.W. Norcross, J. Phys. B 5, 1482 (1972).

3. J. Mitroy, I.E. McCarthy and A.T. Stelbovics, J. Phys. B 20, 4827, (1987).

4. P.J.O. Teubner, J.L. Riley, M.J. Brunger and S.J. Buckman, J. Phys. B 19, 3313 (1986).

5. E.A. Enemark and A. Gallagher, Phys. Rev. A 6, 192 (1972).

6. J.L. Riley, P.J.O. Teubner and M.J. Brunger, Phys. Rev. A 31, 1959 (1985).

7. H.W. Hermann, I.V. Hertel, W. Reiland, A. Stamatovic and W. Stoll, J. Phys. B 10, 251 (1977).

8. J.L. Riley, Ph.D. Thesis, The Flinders University of South Australia (1984) - unpubhshed.

9. A. Fischer and I.V. Hertel, Z. Phys. A 304, 103 (1982).

10. N. Andersen, J.W. Gallagher and I.V. Hertel, Phys. Rep. 165, 1 (1988).

11. J.V. Kennedy, V.P. Myerscough and M.R.C. McDowell, J. Phys. B 10, 3759 (1977).

12. H.W. Hermann and I.V. Hertel, Comments At. Mol. Phys. 12, 127 (1982).
51

Theoretical interpretation of the associative ionization reaction between laser excited

sodium atoms : Atomic and Molecular points of view

Anne HENRIET+ and Frangoise MASNOU-SEEUWS

Laboratoire des Collisions Atomiques et Moleculaires, Bat. 351, Universite Paris-Sud

91405 ORSAY Cedex France

The experiments on associative ionization between two laser excited alkali atoms have demonstrated that the ion signal

markedly depends upon the angle between the atomic beam and the polarization vector of the exciting laser light. In the atomic point of view, first developped by Nienhuis, this angular dependence is analyzed to determine which initial preparation of the two atoms is the most favourable. In the molecular point of view, the problem of the autoionization of

the Na2 molecule is treated so as to know which molecular symmetry is the most favourable one. The first order MQDT

treatment proposed by Giusti (1980) for dissociative recombination is applied to the present problem. Two-electron

calculations for the intermediate Rydberg states of Na2 are used to define a diabalic representation, where dissociative doubly

excited states Qj are interacting with mono-excited Rydberg series, converging to the Na2 ground state X. The ionization probability is controlled by the strength of this electronic coupling and by the position of the crossing between the Qj and

the X curves. The cross sections may thus differ considerably from one molecular symmetry to another, explaining the strong anisotropy observed experimentally. A link between both points of view has been established by Jones and Dahler

who considered a sudden change from atomic to molecular "Hund's case a" representation at infinite internuclear distances. Although some aspects of the problem still have to be clarified, the conclusion is that the associative ionization reaction is more likely to proceed when the two atoms are prepared in Zeeman sublevels j=3/2, mj=l/2 or -1/2 and form a molecular

state of symmetry.

example for the formation of a chemical bond, the associative ionization reaction presents a fundamental

interest. In the initial state, all the electrons are bound

while the relative motion of the two atoms is free ; in the

final state the vibrational motion of the two nuclei is bound while a free outgoing electron is taking off the excess energy and eventually some part of the angular

momentum.The major difficulty lies in the treatment of

bound-continuum interactions which involves a very large

number of molecular states. However, the strong dependence of the observed

ion signal as a function of the initial preparation of the two

atoms indicates that some molecular symmetries should be more favourable than the others for the molecular

autoionization process. In order to analyze this symmetry

effect, one can adopt two points of view : 1) An "atomic" point of view, first developped by

Nienhuis"^, analyses the experimental signal so as to determine the relative value of the cross-sections for a given

initial state, when the two atoms are prepared in particular

Zeeman sublevels.The question therefore is : in which

Zeeman sublevels should we prepare the two atoms in order

to get the maximum signal ? 2) A molecular point of view, in which the autoionization of the Na2 molecule is considered. Among

the twelve molecular states which are correlated to the Na(3p)4-Na(3p) dissociation limit, ten may contribute to the reaction (1) (the two others being Z' states, a symmetry

which is not possible for the final state). The question then

is : for which of those ten states is the autoionization process most probable ?

iptrftduct'on

During the past few years, many experimental results have become available for the associative ionization (AI)

reaction between two laser excited alkali atoms, in particular for the reaction Na(3p) -I- Na(3p) ^NaJ-i- e (1) The results can be summarized as follows : l)The value of the total cross-section is of the order of 1 at room temperature, that is more than one

order of magnitude smaller than the geometrical crosssection : the minimum of the Na2 ground state potential curve being located around 7 a.u, one should expect a geometrical cross-section of 43 A2• An unexpectedly large increase of the cross-section is apparently observed at ultra- cold temperatures-^.

' 2)A strong anisotropy effect is observed^ when

the two atoms are excited using polarized laser light. The

ion signal can vary by a factor of two when the polarization vector of the laser is turned by 90°. Detailed results concerning the sodium measurements using linearly or

cicularly polarised light are presented in the following talk by H. A.J. Meijer. It is remarkable that a similar conclusion is obtained^ when two potassium atoms are

considered.3)In the latter case however, a competition with

the K"'"K' exit channel is observed. The interpretation of such experiments is a challenge for the theoreticians : being a particularly simple
52

The link between the atomic and molecular points of view then needs to be established. In the work of Jones and Dahler9, this is done by assuming a sudden change of

coupling at large internuclear distances from the I j mj >

atomic states to the molecular states corresponding to Hund's case "a".

2. Atomic point of view : analysis of the

measurements in terms of Nienhiiis' theory.

Let us consider a collision event where in the initial state two atoms are prepared in Zeeman sublevels Iji mi>lj2 m2> and collide with relative velocity Vi and

angular momentum J. In the final state, a Na2 molecular

ion is formed in a ro-vibrational level lv,J'> while an

electron escapes with a final velocity Vf

[Na(3p ji mi) + Na(3pj2 m2)] (Vi,J)

NaJ(v,J')+ e( Vf ) (2) We shall call : I a > = Iji m i> I j2 m2 > the initial state of the two atoms and f( a,Vi ,J v,J',Vf) the

probability amplitude for this collision.. Let us consider an experiment where the angle between

the polarization axis of the laser light (direction of

polarization for a radiation, propagation direction for n

radiation) and the atomic beam is 9. The number of ions

detected per unit time, R(G), can be computed from the trace of the the product of two matrices : R(e) = Tr[G.p] (3) where p is the product of the density matrices of the two

atoms and depends both upon the polarization of the ligth

(linear, circular) and upon the angle 9. G is a detection

operator defined as

<a I G I a'>= I ^lf Vf/ui X Z (4)

I J' V

J{f(a,Vi,J ^ v,J',Vf) f*(a',Vi,J ->v,J',Vf)d(Vf/IVfl)}av

In (4), |af and \ii are the reduced masses in the final and

in the initial state respectively, the average is performed

over the distribution of the initial relative velocities, the summation is over all possible rotational and vibrational

states of the final molecular ion and the integration is over

the direction of the velocity of the ejected electron.The

diagonal elements of G are simply rate constants for the formation of a molecular ion starting from two atoms

prepared in a given combination of Zeeman sublevels /a />

. The non diagonal matrix elements are less obvious

coherence terms corresponding to an interference between

the contributions to the ionization process starting from

two different initial states I a > and I a'> of the two atoms. By expanding the two matrices p and G in spherical tensors with appropriate quantization axis, Nienhuis finds

that R(9) can be expressed as a combination of a limited number of independent parameters. For instance, in the case of the excitation of the F=2^3 hyperfine component of

the D2 line with linearly polarized light, R(9) has only

three Fourier components :

R(9)= Ro+ Rl cos(9) +R2 cos(29 ) (5)

depending upon five independent parameters, among which

three are combinations of the diagonal matrix elements of

the G matrix and two are coherence terms. Assuming that the latter may be neglected, it is possible to determine from

experiment three quantities A, B, C : A=l/2 [K(3/2,3/2) +K(3/2,-3/2 )] B=l/2 [K(3/2,l/2) +K(3/2;-l/2 )] (6) C=l/2 [K(l/2;l/2) +K(l/2;-l/2 )] where the rate constant K(mi;m2 ) is defined as : K(mi,m2 ) = <ji=3/2,mi ; j2=3/2,m2 I G Iji = 3/2,mi ; J2 = 3/2,m2 > (7) Both in the case of Na(3p)+Na(3p)4 and K(4p)+K(4p)6

collisions the Fourier analysis of the signal indicates that the associative ionization reaction is favoured when the two atoms are prepared in Zeeman sublevels with mj

=+1/2 or -1/2. However, as will be discussed by H .Meijer, the coherence terms should not be neglected.

More information about the collision process can be

obtained by varying the polarization of the exciting light : the Fourier analysis of the ion signal, in the most general

case, contains also terms in cos(39). The results are discussed by H.Meijer. Future work should discuss the validity of the

representation of the operators in a subspace restricted to the 6x6 sublevels of the two 3p atomic configurations : calculations by Kowalczyk^^ indeed predict a long range (R=30 a.u) avoided crossing between the potential curves

correlated to Na(3p)+Na(3p) and one potential curve

correlated to Na(3s)+Na(4d).

3. From atomic to molecular point of view : analysis of the measurements in terms of .Tones and Dahler' theory.

A simple model for the formation of molecular orbitals from atomic orbitals consists in assuming a sudden

decoupling of the spin of the two atomic valence electrons

at infinite internuclear distance, the orbital momentum

being then coupled to the internuclear axis. The atomic

states with mj=3/2 or -3/2 will then give k orbitals, while

the states with mj=l/2 or -1/2 will lead to a and n orbitals.

Ij mj> II mi s ms >

13/2 3/2>^ |1 1 1/2 1/2 > (8)

13/2 l/2>^ V2/3 |1 0 1/2 1/2 > + Vl/3 |1 1 1/2 -1/2 >

It is then easy to predict which molecular states are formed for an initial preparation of the two atoms in particular Zeeman sublevels. This is done in Table 1, extracted from ref 1 1
53

Table 1. Population of the various molecular states from the atomic states assuming a sudden change of

coupling from atomic to molecular Hund's case a. mi

m2

3a,

(|-)3nu

(i)3ng (i)3nu

(i)in„

m' g

12

(|)^2^(a2) (|3z;(a2) (|)3n.

(|)3nu

(-)^na

(^)3lg(rc2)

(|-)3nu (^)1i^(k2)

(|)3n.

(|-)^ng

(J) 'ng

12 i)^ng

(|-)inu

(^)3nu (|) 3Au

32

3Au

For some symmetries two states are correlated to Na(3p)+Na(3p), one formed with two a orbitals and the other witht two k orbitals. It appears clearly that the molecular states of ^ symmetry constructed with two O

orbitals can be formed only when the two atoms are prepared with a magnetic quantum number I mj j- I m2 I =112.

In the work of Jones and Dahlcr^, the ion signal

R(9) is determined as a sum from the contributions Ri(6) of

the various symmetries weighted by the population of each

molecular state li>. Such populations ai are easily

determined from the matrix density p by the transformation

described above. The coherence terms are neglected.

RO) = Z ai Ri(e) (9)

i The angular dependance of the various Ri(6) is analyzed, the contribution of A states being neglected. The conclusion is that the experimental results for the Na(3p) + Na(3p)

associative ionization process can be interpreted as due to a dominant contribution of the symmetry together with

a weaker contribution of the ^^g^ (^^^ > ^ ^u'^

molecular symmetries. Future work should also consider the A states, which according to the experimental work of Wang et al^2

are contributing to the reaction. We shall now discuss the determination of the autoionization probability for the various molecular

potential curves

.

4. Molecular point of view : treatment of the

autoionization of the Na"> molecule

One of the main difficulties in treating the associative

ionization problem is the large number of channels that are involved. We know that of the twelve molecular potential curves correlated to the Na(3p) + Na(3p) dissociation limit, ten may contribute to the reaction (1). Such curves are embedded in the Rydberg series converging to the various Na(3s) + Na(nl) limits : the (3s + 5s) and (3s + 4d) limits are respectively 92 meV under and 70 meV above the doubly excited limit. Turning now to the reaction region,

the full treatment of the autoionization problem for a given

molecular symmetry requires in principle the consideration of an infinite number of states, corresponding to molecular

Rydberg series We shall see that this complexity can greatly be reduced by use of a multichannel

quantum defect (MQDT) approach ^3 We first compute the potential energy curves of the Na2

molecule : as the final state of the reaction (1) involves

Na2, it is also necessary to describe the molecular ion. Due

to the weak bonding energy of the sodium valence electron compared to the core electrons, a model potential treatment should be satisfactory The Na2 ion is then treated as an

effective one-electron system, and the Na2 molecule as an

effective two-electron system. We treat the core-polarization

effects via effective operators the interaction between one

electron and one core being represented by a parametric

potential fitted to the sodium experimental spectrum 16

Fine structure effects are neglected in this treatment.

In a first step, the Na2 orbitals Xp are evaluated

accurately by solving the one electron Schrodinger equation
54

in a space defined by a large basis set of generalized Slater

orbitals in prolate spheroidal coordinates^^. The potentiiil curves for the ground and first excited states of the Na'\

molecular ion are displayed in Fig. 1. E{au)

- Na^Na(3p)

Na%Na(3s)

10 15 R(auf

Figure 1. Potential curves for Na2-

We next solve the Na2 problem by expanding the two

electron wavefunction in antisymmetrized products of two

Na2 orbitals Xp and Xq

f(l,2) = Zai [xl(l)x'q(2)±xl(2)xlw} (H)

We have chosen to perform a numerical evaluation of

the two-electron integrals^ ^. An expansion on 30

configurations for the I symmetry, 20 for Uie k symmetrj',

is sufficient to obtain, by diagonalization of the full hamiltonian 3i(l,2,R), the various adiabatic potential curves of Na2^^ with an accuracy which can be assessed by

comparing to spectroscopic data. The conclusion is that the energies are presently determined to within 200 cm'^

(24 meV) and the position of the minima to within 0.1 A. Moreover, an independent check of the calculations is the good agreement obtained with the potential curves computed by Jeung^O up to the 3s + 5s dissociation limit using an ab initio pseudo-potential treatment with a gaussian orbitals basis set. As some adiabatic potential curves display many avoided

crossings, the Rydberg scries being strongly perturbed by a doubly excited state, it appears more convenient to use a

diabatic representation . The procedure, first introduced by O'Malley^^ to treat autoionizing atomic states, consists in diagonalizing the two electron hamiltonian Jl(l,2,R) by dividing the configuration space, for a given molecular symmetry, into two subspaces :

i) in one subspace, hereafter named V, we consider

configurations such that one electron stays on the bonding

jg3s Na2 orbital. Partial diagonalization in P yields

regular Rydberg series converging to the Na2 ground state X. For all the states I P j > of a series k, the ionization energies are obtained by the Rydberg formula : <P^ |H(1,2,R) |p^>-Ex(R) =

5ijR/(n^-Uk(R))2 (12)

In (12), Ex (R) is the energy of the Na2 ground state,

is the Rydberg constant, n^ an integer number increasing

by one unit from the state I i > to the next upper one, and

|j.k(R) is indeed independent of the index i. Such states correspond co a Rydberg electron moving in the field of a ground state Na2 core.

ii) in another subspace, hereafter named Q, we consider

doubly excited configurations, such that both electrons occupy excited Na2 orbitals. Partial diagonalization in Q

yields doubly excited potential curves, corresponding to Rydberg series converging to an excited state of Na2 or more generally to a linear combination of such excited

states. In order to avoid an incorrect dissociation, we have

modified the subspace Q into Q ' by orthogonalization to the lowest state obtained in (i). For most symmetries, this procedure causes a minor modification of the Qj states. An

example of such states is displayed on Fig.2. for the ^Ij^

symmetry.

E (au) '

-0.15

-0.20

-0.25

Na(3p)+Na(3p)

5 iOR(au)

Figure 2. Diabatic potential curves for Na2 ^^ydashed lines, "p" Rydberg series ; solid lines "f Rydberg
55

series ; dash-dotted line, doubly excited states. The

shadowed curves correspond to the ground and first excited

states of Na2 .

After this block diagonalization, the electronic hamiltonian

H (1,2JR) has non zero matrix elements between a state of

P and a state of Q . As the states of P are grouped in Rydberg series, there exists^^ a scaling law for such matrix

elements

:

<I* i H ( 1,2,R ) I Qj > = (n^ - Uk(R))-3/2 Vjk (R) (13)

where the reduced interaction VjkCR) may be considered as a

constant along a Rydberg series. Examples for this scaling

law are displayed in table 2. We should note that the scaling

law is no longer valid for R>10 a.u when the two atoms

begin to part and that the reduced interaction is a slowly

varying function of the intemuclear distance. Table 2. Example of a scaling law for the interaction of

the second doubly excited state of symmetry with two

levels of a Rydberg series in the mono-excited space P

R = 5 ni- |ii(R) 3.90 4.90

R = 6

Vjk(R)

ni- ni(R)

0.048

3.88

0.046

4.89

R = 7 Vjk(R)

ni- w(R)

0.044

3.80

0.042

4.81

R = 8 Vjk(R)

ni- w(R)

0.049

3.78

0.044

4.78

R = 10

l^jk(R)

ni- \ii(R)

0.048

3.63

0.043

4.66

0.049 0.037

The associative ionization can therefore be understood as a two electron process, where in the initial state one

electron occupies an excited antibonding Na2 orbital, the oth^ electron occupying a Rydberg orbital. The transfer of

the inner electron towards the ground state Na2 orbital yields enough energy (>2eV at equilibrium) to enable the second electron to escape. We have considered separately two groups of doubly

excited states, hereafter labelled A and B. The group A

contains the second doubly excited state of each symmetry

which, for intemuclear distances R > 10 a.u., is mainly

described by configurations containing two of the four

orbitals correlated to Na''"+ Na (3p). The group B contains

the first excited state of each symmetry, which for R > 10

a.u. is mainly described by a configuration involving the au3s orbital. The corresponding potential curves are displayed on Fig.3 and 4 respectively.

E(au)

Figure 3 Doubly excited states of Na2 correlated to the 3p+3p dissociation. limit.

Na(3p)*Na(3p)

Figure 4. Doubly excited states of Na2 corresponding for each symmetry to the lowest state in the subspace Q. They

are correlated at infinity to an ionic curve. The '^'Zt curve is lower and not visible on the figure.

Following Giusti^^ the autoionization of a molecule

can be described in the framework of a MQDT treatment. Let us consider the ionization of the doubly excited state Qj

resulting into a continuum state corresponding to the extrapolation of the Rydberg series k.

(Na2**){-^(N4(x2Ej),vJ) + e(e))k (14)

We assume that the angular momentum ^iVj(J+1) of

the heavy particule motion is conserved. The corresponding
56

cross-section may be evaluated, in the weak coupling

approximation, by : J max

a^^(E) = ^r I (2J+l)aj|,(E) 2^ (15)

In Eq(15), E is the collision energy, r the multiplicity ratio between the final and the initial state, J the partial wave and

T ajjj (E) a dimensionless quantity such that

vmax 4 | ti\ \ 2

(16) v=0

w

^'j = ^ l?CvJ (R) Vjk(R) 4

(E,R) a R (17)

In (16), the summation is over the vibrational levels of

Na2 which may be populated through the reaction (14) at

energy E. In (17), Xy is the vibrational wavefunction in the

final state, yjk(R) the reduced interaction defined in the Eq(13) and ^ (EJl) the radial wavefunction associated to the

relative motion of the two atoms interacting with the

potential Qj(R) for given energy E and angular momentum

1). As Vjk(R) is a slowly varying function of R

(see table 2) the autoionization process finally depends of

quantities which can be estimated as :

4j = ^ % IXvJ (R) 4

(E,R) dR (18)

In (18) yjk is an average value and the integral is merely the overlap between the radial wavefunctions

describing the nuclear motion in the initial and in the final

jtate.The autoionization probability is thus controlled both

by the strength of the electronic coupling and by the Franck-Condon factor for the initial and final nuclear

wavefunctions. Such factors depend markedly upon the

relative position of the Na2 ground state potential curve X(R) and of the doubly excited curve Qj(R). We have

reported in Fig. 5 the vibrational wave functions XvJ for various levels of the molecular ion together with

continuum functions fj (E,R) computed at 0.07 eV both for crossing and for non crossing potential curves. Even in the case of a strong electronic coupling, the autoionization process can hardly take place for any value of the

vibrational quantum number v when the two curves are not crossing. It is also negligible for the vibrational numbers

v < 3 when the crossing is located at intemuclear distances R > 8 a.u. We may therefore predict a strong selectivity of the autoionization process at low collision energy E in favour

of molecular symmetries leading to a doubly excited

diabatic curve presenting a crossing with the NaJ ground

state potential curve close to the minimum.

.V"V = 2

8 R(au)

Figure 5. Wavefunctions for the nuclear motion in the

final state (vibrational functions Xy computed for J=0) and

in the initial state in the case of 1) a non crossing curve

Qj('R) 2) the crossing curve Qa discussed in text.

Considering the group A of states, displayed in Fig.3, we see that one of the tv/o -^Z^ curves, hereafter labelled Qa, crosses the X curve at R = 7.5 a.u. A curve,

hereafter labelled Qb, is crossing the X curve at R = 8.1

a.u. The autoionization via a ^riu state necessitates a

collision energy E > 0.14 eV. Therefore we may conclude

that for E < 0.14 eV, autoionization proceeds preferentially via E molecular states. For E < 0.054 eV, the Qa ^S^J" channel is the only possible one. Such a conclusion is in agreement with the experimental findings'^'^, and with the analysis of Jones and Dahler^ (see above), provided that we may assume an adiabatic correlation of the Qa state to a

state of the separated atoms with two a orbitals. The

states of the group A are constructed mainly with the (ag

3p au 3p) and (Kg 3p Jiy 3p) configurations. There is a crossing at R = 13 a.u between the energies of those two

configurations, the first one being lower at large distances. The comparison with the experimental results could indicate

that this crossing is adiabatic. However, a correct treatment of the population sharing from the states of the separated atoms to the molecular states has to be considered. In the case of the Izj" curve, the discussion is even more
57

complicated, as four configurations (ag 3pr (cfu 3p)'^ (ng

3p)-^ and (ruu 3p)2 should be considered. Assuming that in the group A of states only the -^Z^ Qj^

state can be populated, and that there is a statistical 3 population of the Qa state of from the Na(3p) + Na(3p)

4

limit, the r factor in (15) being j, we find a cross section

of 12 A2 at 0.03 eV, 18 at 0.05 eV, while the experimental cross section^ is of the order of 1 A^. In our calculations Jmax is the maximum rotational number

for a given value of v and we have considered an arbitrary

extrapolation of the Qa curve to the separated atoms. The

presence of a small long-range barrier would change the

results. We also predict a selective population of the levels v=2

and v=3. Turning now to the group B of states (see Fig.4) we

see that provided they may be populated from the Na(3p) f Na(3p) dissociation limit, the ^Ag, -^Au, ^Tl^ and -^11^ doubly excited states could also contribute to the ionization process. For instance the ^ Ag state is mainly represented by

the (7Cg3p 7tu3p) configuration at short distances, which has

a smaller energy than the (au3s 5g3d) configuration.

Although this situation is reversed at large distances, we

may note that the states of the group B should be correlated

to ionic Na'''-Na' states, the Na' ion being excited into II jnd A states. A detailed treatment of the correlation between the atomic representation at large R and the diabatic molecular

-•epresentation at small R is therefore required

5. Discussion

It is thus possible to treat the autoionization of the Na2

molecule by considering the interaction of dissociative doubly excited states with mono-excited Rydberg series. Several points of this treatment still need to be refined :

i) the potential curves are required to better accuracy. Ab initio calculations by W. Meyer and colleagues in Kaiserslautem offer hope for an improvement in the future.

ii) the MQDT evaluation of the K matrix should l)e extended to second order, including the indirect ionization process via vibrational autoionization of a mono-excit(5d

Rydberg state Na(3p)+Na(3p)-^(Na2)**->Na^(viJ) ^ NaJ (v,J) -1- e

^ (19)

Such work is presently in progress. Nevertheless, the qualitative conclusion that a -^Z^ doubly excited state is mainly responsible for the auto- ionization process is consistent with the Nienhuis'

interpretation of the angular dependence of the experimental

ion signal showing that, for coUision energies E = 0.1-0.3 eV, the reaction (1) is favoured when both atoms are

i3 1 prepared in a I t :r > Zeeman sublevel. It also agrees wi th

the conclusion of Jones and Dahler^ who assume a sudden

change of coupling at infinite intemuclcar distances. At lower collision energies, the hypothesis of a sudden

decoupling of the spin is no longer justified and a correct treatment of the connection between the atomic state representation in the asymptotic region and the diabatic molecular representation in liie autoionization region should

be realized in a nearfuture.

^Present address : Laboratoire des interactions ioniques et moleculaires. Universite d'Aix-Marseille 1 - Centre Saint Jerome, 13397 MARSEILLE Cedex 13

Acknowledgements.

The authors wish to acknowledge for the stimulating

collaboration with A Giusli-Suzor and O.Dulieu and the discussions with N.Andersen, H.A.J. Meijer,

R.Morgenstem, J. Weiner and R. Mc Carroll. References.

1. J. Huennekens and A. Gallagher, Phys.Rev.A28. 1276

(1983)

2. R. Bonanno, J. Boulmer and J. Weiner, Phys.A 28, 604

(1983) R. Bonanno, J. Boulmer and J. Weiner, Comments

At.Mol.Phys. 16, 109 (1985)

3. P.L. Gould, P.P. Lett, P.S.Julienne, W.P.Phillips,

H.R.Thorsheim and J.Weiner, Phys.Rev.Lett.M, 788

(1988)

4. J.G. Kircz, R. Morgenstern, and G. Nienhuis,

Phys.Rev.Lett. 48, 610 (1982)

5. M.X. Wang, J. Keller, J. Boulmer and J. Weiner

Phys.Rev.A M, 4497 (1986)

6. L.Brencher, B.Nawracala and H.Pauly, Z.Phys.D.lO.

211(1988)

7. G.Nienhuis, Phys.Rev.A26,3 137(1982) and XIV

ICPEAC invited papers

8. A.Henriet and F.Masnou-Seeuws submitted to Z.Phys.D.(1989)

9. D.M. Jones and J.S Dahler, Phvs.Rev.A35, 3688 (1987)

10. P. Kowalczyk J.Phys.B.17 817 (1984)

11. A.Henriet ,thesis,Orsay, unpublished (1988)

12. M.X. Wang, J. Keller, J. Boulmer and J. Weiner

Phys.Rev.A35 934 (1987)

13. A. Giusti J.Phys.B 13 3867 (1980)

14. A Dalgamo, Atomic physics, vol.4 (Plenum Press, New

York 1975) pp 325-334

15. C Bottcher and A. Dalgamo, Proc. Roy Soc Lond. A340

187 (1974)

16. M. Klapisch, Cornput.Phys.Commun.2 239 (1971)

17. A. Henriet J.Phys.B.18 3085 (1985)

18. A. Henriet, C. Le Sech and F. Masnou-Seeuws,

Chem.Phys.Lett.158 389 (1989)

19. A.Henriet and F. Masnou-Seeuws J.Phys.B.2i L339

(1988) and J.Phys.B. in press (1989)

20. G.H.J. Jeung Phys.Rev.A35 26 (1987)

21. T. O'Malley Adv.At.Mol.Phys. 7 223 (1971)

22. M.J. Seaton Rep.Prog.Phys. 46 167 (1983)
58

Detailed study of polarization effects on the Associative Ionization

in Na(3p) - Na(3p) collisions

H.A.J. Meijer

Fachbereich Physik, Universitat Kaiserslautern, FRG

1) Introduction

In recent years many groups have investigated associative ionization occurring in

(thermal) collisions between two laser

excited Na(3p) atoms:

Na(3p) + Na(3p) - Nao* + e" (1)

which is one of the possible reactions

that occur during such a collision. Fig.

1 shows schematically a few of the molec0

Internijclear distance Fig. 1 Some potential curves for Ha^

ular curves of the Na dimer and their

dissociation limits. This limit for 3p +

3p lies just above the minimum of the

Na2* potential. Kircz et al. [1] were the

first to show that the cross section for

(1) depends strongly on the polarization

of the collision partners. After this

publication, several groups investigated

the polarization, as well as the collision velocity dependence [2,3,4]. Aim of

this study is twofold:

First, we want to produce many accurate

and mutually consistent data on process

(1) , so that comparison with theory is

meaningful

Second, we want to demonstrate a complete

experiment, in the sense that in the given geometry of the experiment the maximum

amount of information becomes available.

In order to achieve these goals, a highly

automatized apparatus has been developed,

in which many checks have been incorporated, especially concerning the laser

and atomic polarization, and the laser

intensity. To describe the experimental

results, we use the theoretical analysis

of Nienhuis [5] , which formulates the ex- perimental results in terms of atomic

states, i.e. as a function of the initially prepared situation. No assumptions

are made yet about the transition from

atomic to molecular states.

2) Experiment

The experimental setup is shown schematically in fig. 2. Two counterrunning thermal beams of Na are intersected at nearly

right angles (87''') by light from a cw dye

laser, tuned to the F, = 2 - F =3 hy-

/ u perfine component of the Na(3-Si/2

3-P3/2) transition. The laser beam is ex- panded and directed through a rectangular

diaphragm, resulting in a spatially homogeneous laser spot of (hxw) 9x5 mm. After it has traversed the Na beams the

laser beam is directed into its own path.

The rectangular laser spot defines the

interaction region, where ions can be

formed according to process (1) . The

total Na density in the interaction re- gion is about 1-10'^ cm-^. All ions

created in the interaction region are extracted by a weak electric field and

are counted by a high current channeltron. The fluorescence light is monitored

by five photodiodes at various angles,

allowing us to keep track of the atomic

density and the atomic polarization (see.

mirror

beam Fig. 2 Sckemat ic overview of the apparatus
59

2.2). The laser intensity has been

chosen only 8 ± 2 mW/cm^, i.e. far from

saturation, in order to keep the

so-called trapping effect caused by the

hyperfine splitting of Na as small as possible. Such a trapping effect would

introduce a spurious velocity dependence

of the cross section and, even worse,

could influence the prepared polarization

of the excited atoms significantly [7,8].

The magnetic field in the collision

center can be varied between 0 ± 5 mG and

> 2 G, in arbitrary direction. Normally

it is kept 0 mG {i.e. the earth's magnetic field compensated) , since magnetic

fields influence the atomic polarization.

However, we can also deliberately introduce this influence, thereby having to our disposal an additional tool to vary

the polarization [7]

.

We excite atomic beam 2 by the incoming

laser, and atomic beam 1 by the reflected

one, using the Doppler shift introduced

by the crossing angle of 87"^. By varying

the laser (detuning) frequency we can select the velocity class of atoms excited in both atomic beams, and thus the

collision velocity of excited atoms. This

method requires an excellent long-time

frequency stabilization, which is performed in a separate apparatus using a third laser [9] . Furthermore, the selective excitation of

the atomic beams 2 and 1 by the direct

and reflected laser beam, respectively,

implies that the polarization of atoms in

beam 1 can be made different from the

polarization of those in beam 2, simply

by changing the laser polarization of the

reflected beam. To take full advantage of

this possibility we use in the incident

laser beam a rotatable \ll plate in com- bination with a polaroid, to rotate the

polarization vector. Also, a ^/4 plate

can be installed, in order to prepare

circularly polarized light. For changing

the polarization of the reflected beam we use either a rotatable /I/4 plate or a rotatable J/8 plate. Since the laser beam

passes these retardation plates twice

they act as a \!1 and a /4/4 plate, re- spectively.

By using the appropriate combination of

these retardation plates, we can choose

either linear polarization with the polarization vector at arbitrary direction.
60

or circular polarization with arbitrary

helicity, and this independently for the

direct and the reflected beam.

The experimental setup is completely

automatized. Fig. 3 shows an overview.

The photodiodes and the ion counting

system are read, and the magnetic fields

are set automatically. Furthermore, a "laser okay" signal, coming from the

laser stabilization apparatus, is interpreted : measurements are stopped during

the time this signal is "not okay". All

retardation plates are mounted on com- puter-driven stepper motors; two of them

can be moved in and out of the beam by means of DC-motor driven translators,

again computer controlled. Finally, also

the shutters for the atomic beams and the

reflected laser beam are computer operated. They are needed to separate the ion

signal resulting from a collision between

atoms from different beams (so-called

head-head collisions) from the signals

from collisions between atoms within one beam (head-tail collisions). Furthermore,

they allow us to measure the fluorescence

light (and thus the relative density and

the polarization) of the two atomic beams

separately, and finally we can correct

for the small "cross-over" excitation

(i.e. the direct (reflected) laser beam

exciting a small amount of the atoms from

beam 1 (2) ) . Two very crucial points in the experiment

need further explanation:

2.1) The laser intensity stabilization

Apart from small, random fluctuations in

the laser intensity, which increase only

the random noise in the measurements, the

rotation of one or more retardation

plates in the beam also introduces a systematical angle-dependent variation in

"Fc V)

the laser intensity, and thus in the ion

signal (which is quadratically dependent

on the intensity!). Since we are looking

for angle-dependent variation in the ion

signal caused by polarization dependence

of reaction (1) , it is clear that this

effect could influence the measurements

to a large extent. Therefore we developed

a computer operated intensity stabilization system, shown in fig. 4. It is based

on a Pockels cell in combination with a polarizer. In order to use this device as

a stabilization system, it needs a feedback signal. However, in our system with

a direct and a reflected laser beam, and

a variable polarization, this signal has

to be realized in a complicated way. Two

photodiodes on the sides of the slit

diaphragm monitor the laser light intensity. Still, their signal is not suited

for stabilization since a rotation of the

\ll plate varies the intensity in the

center of the spot different from that at

the sides. Furthermore, variations in the

reflected beam would not be accounted for

at all.

Our solution is the following: Before the

start of a measurement at a certain position of the retardation plates, another

photodiode is shifted automatically into

the beam, just in front of the mirror.

Our computer measures this signal and it

adjusts this signal to a certain preselected value (corresponding to the irradiance desired in the experiment) , by

changing the reference signal for the

Pockels cell. The feedback system itself

will keep the signal of the photodiodes

on the slit equal to that reference signal, thereby compensating for random

fluctuations in the laser intensity

during a measurement. The photodiode in

front of the mirror is shifted out of the

beam again, and the measurement itself

X/4 OR \/3 PLATE MIRROR

PHOTODIODE

fiQ'. 4 The laser intensity stabilization sysien.
61

starts. After finishing the measurements

at that certain position of the retardation plates the photodiode is moved into

the beam again and checks the intensity.

If the check is satisfactory the retardation plates are rotated to a new position

and the procedure starts again.

2.2) Atomic polarization and density of

excited atoms

In order to analyze the data we need to

know the relative density of atoms. Furthermore we want to check the atomic polarization. In principle, the emitted

fluorescence light contains this information:

(n. +n_. ) (l+cos2(7>) + 2n^sin2i25

I,= Tin , ^ ^ \ (2)

f ' e (n^+nQ+n_^)

with 7} the overall detection efficiency,

n^ the total density of excited atoms,

and n_^,nQ and n^ the density of atoms in

the magnetic sublevels M = -1,0,1, re- spectively. Finally, j is the angle

between the detector and the polarization

direction. Using our five photodiodes, we

are able to measure i/n^, the relative

density of excited atoms during one measurement series, as well as the ratio

(n_^+n^)/nQ. The latter was compared to

the theoretically expected value. In some cases there were significant deviations,

which were caused by a bad polarization

quality of the reflected laser beam, (in

fact the polarization quality of the

reflected laser beam can be checked only

in this indirect, but very reliable way.)

Different measurement series have in

principle somewhat different Tj's. Therefore, every measurement series must contain several measurements, at which both

atoms are excited with linearly polarized

light, at an angle $=0^ . Comparison of

these measurements learnt that 7} varied

by no more than ±10%. After correcting

for these variations, the measurement

series are now comparable to each other

within 5% (for more details see [7,8]).

3) THEORY

The process of associative ionization can

be characterized by ionization amplitudes

f(M^Mg) which depend on the magnetic

quantum numbers and Mg of the colliding atoms. Absolute square values of

these amplitudes represent cross sections

(T(M^Mg) = !f(M^Mg)|2, and the observed

ion rate can be written (in a somewhat

simplified form)

Vk ^ ^ ^ ^ "a"a ^b'^b

where , and p , are elements of the

a' A B'

density matrix p of the excited atoms A

and B. We define the detection operator

G:

(4)

This G contains all information that can

be extracted out of a measurement with

this specific geometry. For each basis of

M states the diagonal elements of G are connected to cross sections (r(M M ) , A B whereas the off-diagonal ones represent

coherence terras, connected to ionization

amplitude products f (M M ) f

(m:m: ) for

which M, ^ M' or M_ ^ M' However, there

A ' A B ' B is no principal difference between this

contributions, and they convert into each

other if one selects another basis of

atomic M-states. We chose our basis such

that the z-axis of the coordinate frame

is the internuclear axis. This makes our results easier comparable to theoretical

potential curves. The elaborate computations following from this choice can be

much reduced by expansion of the density

matrices and of G in spherical tensors

[5,7,8]. The analysis shows that for our geometry there are 8 independent contributions ( 4 cross sections and 4 coherence terms) to the ion signal if we assume that the influence of the electron

spin can be neglected (L-picture) . If not

(J-picture) , this number is doubled (5 cross sections and 10 coherence terms)

The maximum amount of independent parameters we can deduce out of our experiments

with arbitrary combinations of linearly

and circularly polarized light is 7. However, if we use the magnetic field as an additional tool for influsneing the atomic polarization, we can increase this
62

number to 8, The measurement types we have to perform are indicated in fig. 5,

together with the expected dependence of

the ion signal on the polarization angle

9 (linearly polarized light) and/or on

the helicity (+/-)

casa Atom 1 At-om 2 8 - depenCenr rerms

cos 29 CCS 48 sin 29

)(8.-

>(9,-8)

(8,90 ) + - —

Fig. 5 The various detection schemes.

4) MEASUREMENTS AND RESULTS

¥e performed all 7 measurement types,

each at about 15 different collision ve- locities. Figs. 6 and 7 show some exampies (r'^'^'/r'*'-^' and R**'^', respectively) . Fitting the results provides us with an overdetermined set of Fourier

coefficients for all collision velocities. Via a least squares analysis we can

get the results for 7 cross sections and

coherence terms, as well as perform a check on internal consistency [7,8].

Together with the results for the magnetic field dependence [6] we end up with

all 8 cross sections and coherence terms

for the L-picture. However, one cross section is significantly negative, which

means that the L~picture is not valid.

Therefore we have to present our final

results in the J-picture, which we unfor0 90 180 polarization angle (degrees)

Fig. 6 Example of a measurement with

linearly polarized light.

H103* V.

Vpgj -2140 m/s

v,,gj - 910 in/s

-90 90 polarization angle jdegrees)

Fig. 7 Measurements with a combination of

linearly and circularly polari zed light.

tunately could not determine completely.

Fig» 8 shows the three (average) cross

sections. Furthermore we could determine

two coherence terms, both gradually increasing from z 0 arb.un. at 1000 m/s to

33 200 arb.un. at 2200 m/s. (For detailed

information see [7,8],)

5) CONCLUSIONS

5.1) The associative ionization

Our most important conclusion is that the

L-picture description, which has hitherto

been used because of its simplicity, definitely is not valid at collision velocities below 2000 m/s. This means that the

description of the process must involve

the electron spin and/or possibly also

other Na nl states.
63

Comparison with theoretical potential

curves is still preliminary, since important work on them is still in progress

[10,11]. Also the significance of (the

velocity dependence of) some coherence

terms is not yet clear.

0 I L_+-J . 1 . 1 > > 1- 800 1200 1600 2000 2-100 collision velocity (m/s)

Fig. 8 Ike final results for the cross sections, with u- - - {(r + s )/'2

So for the time being, our conclusions

are restricted to what fig. 8 directly

shows, namely that the situation in which

the major axis of the charge cloud points

along the internuclear axis is most favorable for associative ionization in the

collision velocity range we investigated.

Our measurements also provide us with

data at much lower collision velocities:

the "head-tail" signal, taking place at v= 200-400 m/s. Although in this case we

are restricted to measurements at which

both atoms are excited with the same

laser beam and thus with the same polarization, we might still be able to

extract additional information out of

these measurements. This will be done in

a separate publication [12]

.

5.2) The detection scheme

We demonstrated a detection scheme for

possible complete analysis of detection

matrices, which is certainly not re- stricted to our case study of Na(3p) + Na(3p). In fact, the latter system appears not to be the best choice, since in our collision velocity range we must use

the J-picture with more parameters than

we possibly can determine experimentally.

A better case study would be a P- P system, e.g. Ba(6s6p) + Ba{6s6p).

REFERENCES

[I] J.G. Kircz, R. Morgenstern and G.

Nienhuis, Phys. Rev. Lett. 48 (1982), 610.

[2] H.A.J. Meijer, H.P. van der Meulen,

R. Morgenstern, I.V. Hertel, E. Meyer, H. Schmidt and R. Witte, Phys, Rev. A33

(1986) , 1421.

[3] E.W. Rothe, R. Theyunni, CP. Reck

and C.C. Tung, Phys. Rev. A31 (1985), 1362

and A33 (1986) , 1426(E)

.

[4] M.-X. Wang, J. Keller, J. Boulmer and

J. Weiner, Phys. Rev, A35 (1987), 934;

M.-X. Wang, M.S. de Vries and J. Weiner,

Phys. Rev. A33 (1986) , 765.

[5] G. Nienhuis, Phys. Rev. A26 (1982),

3137.

[6] H.A.J. Meijer, Th. Zeegers, T.J.C.

Pelgrim, H.G.M. Heideman and R. Morgenstern, J.Chem.Phys. 90 (1989), 729.

[7] H.A.J. Meijer, T.J.C. Pelgrim, H.G.M.

Heideman, R. Morgenstern and N. Andersen,

J.Chem.Phys. 90 (1989), 738.

[8] H.A.J. Meijer, thesis, Utrecht, 1988

(in English, available on request)

.

[9] H.A.J. Meijer, H.P. van der Meulen,

F. Ditewig, C.J. Wisman and R. Morgenstern, J.Phys.E:Sci.Instrum. 20

(1987) , 305.

[10] A. Henriet, thesis, Orsay, 1988 (in

French); A. Henriet and F. Masnou-Seeuws

,

J. Phys. B 21 (1988), L339; Z.Physrd

(subm.

)

[II] I. Schmidt and W. Meyer, to be

published; I. Schmidt, thesis,

Kaiserslautern, 1989 (in German).

[12] H-A.J. Meijer, to be published.
64

Excitation and Ionization of Atoms — What We Can Learn by Using

Polarized Collision Partners

Klaus Bartschat

Department of Physics and Astronomy, Drake University,

Des Moines, Iowa 50311, U.S.A.

1. Introduction

Recent progress in experimental techniques has enabled very

detailed testing of theoretical models for atomic coUision

processes. The preparation of spin polarized projectile beams

with narrow energy distribution as well as the construction

of sources for polarized target atoms is now a particularly feasible endeavor in several laboratories. Despite this progress, however, the so-called "complete" experiments are

still extremely difficult to perform. This is due to the (often) very large number of independent parameters that have to be

measured with a high degree of accuracy in order to extract the scattering amplitudes from experimental data; in fact, this requires the solution of the nonlinear system of equations

that relates these scattering amplitudes to the individual

observables determined in the experiments.

The question therefore arises what information is really

contained in a given observable and which parameters should

be measured to obtain information about the various effects that determine the outcome of a colUsion process. A

comparison between experimental and theoretical data for these parameters should then allow a profound judgement

about the reliability of the individual theoretical approaches.

In the present work, the above questions are addressed

as follows. We begin with a brief outline of a general density

matrix theory for the scattering of possibly spin polarized

projectiles from also polarized targets. In particular, we

will concentrate on the systematic derivation of the reduced

density matrices which must be used in the description of almost all realistic experiments. Furthermore, some important observables will be introduced, and numerical results for these parameters will be compared with recent experimental

data. In detail, we wiU concentrate on i) the change of an

initial projectile polarization through the scattering process,

ii) the light emitted in atomic transitions after excitation by

polarized electrons, and iii) the ionization of spin polarized

target atoms by also polarized projectiles. A more complete

treatment of the present topic has been given in ref. [1].

2. General Density Matrix Theory

In studying the scattering of spin-^ particles with initial linear momentum kg and spin component ttiq (with regard to a given quantization axis) from a target with total angular

momentum Jq and spin component Mq, we are generally

interested in transitions to a final state with momentum kj

,

spin component m-^ , and a target state with Jj and Mj . The

pure initial and final states are denoted by | jQMQ;kQmo) and

|JiMi;kjmj), respectively. The above transition is described

by scattering amphtudes

/(Mimi;Momo) = {J^M^;k^m^ \ T \ J^M^^'k^m^) (1)

where T is the transition operator. In order to keep the notation as simple as possible, we have omitted the arguments

kg and k^ as well as Jq and Jj in the scattering amplitudes.

We can now write the density matrix of the initial state

as

|JoM^;koTn;)(JoMo;komo| (2)

where the symbol x denotes the direct product. The elements p , andp,^, characterize the preparation density matrix of the projectiles ("p") and the target ("t"), respectively.

Note that by using the direct product, we implicitly assume i an independent preparation of both systems. This is the normal case in practice, but a formal generalization is straightforward. Following Blum [2], the density matrix

elements of the final state are obtained as

f{M[m[;M'^m'^)r{M^m^;M^m^) (3)

where the star denotes the complex conjugate quantity and

j 0 is the scattering angle, i.e., the angle between kg and k^.

| These density matrix elements contain the total information

which can be obtained from the scattering process for a given

set of elements p , Pm'm describing the initial state. The

two sets are related to each other through the scattering

amplitudes, but it should be noted that generally the ele- ments of the outgoing density matrix cannot be factorized

in parts describing the target and the projectile separately

(see also Blum [2], Chapter 3). In the above formulation, a

so-called "complete experiment", i.e., the determination of

all independent scattering amplitudes, is equivalent to the knowledge of all possible density matrix elements of p^^^

for all possible initial states described by Pj„. Apart from

a few exceptions (see, for example, Berger and Kessler [3]),' however, such experiments can hardly be performed in reality.
65

Before going into more details, we point out the main

advantage of the density matrix formalism. It is essentially

the possibility to describe real experimental facts in a sys- tematic way. The initial state in basically every experiment

is described by a mixture of pure states, and in the final state not all possible quantum numbers are determined

simultaneously. While the first experimental restriction is accounted for through the preparation density matrix in eqns. (2) and (3), respectively, a so-called "reduced" density

matrix has to be introduced in the theory to accommodate

the second limitation. Some important examples will be

discussed in the following.

2.1 Generalized STU-Parameters

We will now specialize eqns. (2) and (3) to the scattering

of spin polarized spin-| particles from unpolarized targets. In addition, we will assume that only the polarization of the projectiles is measured after the collision process. In

this case, the initial density matrix elements of the target

are given by p^,^^ = 2J^^m;,m,^ reduced spin

density matrix elements p^, ^ {$) of the scattered projectiles

are obtained by summing over M[ = Mj in eqn. (3). The next step is a detailed analysis of the geometrical and

dynamical symmetries contained in these reduced density

matrix elements, with the final result shown in fig. 1.

If parity conservation is assumed, the above spin density

matrix is described by eight independent parameters — the

absolute differential cross section cr^ and the seven relative parameters Sp, 5^, T^, T^, T^, {/^.^ and U^^. In terms

of these "generalized STU-parameters", the projectile spin

polarization P' after the scattering is given by

(4) which is a generalization of eqn. (3.78) of Kessler [4]. The following geometries are particularly suitable for the

experimental determination of the individual parameters: to begin with, cr^ and Sp can be measured with unpolarized

projectiles while a transverse polarization perpendicular to the scattering plane (P = PyS) is needed to obtain 5^ and Ty.

Finally, the measurement of T^, f/^^,, and U^^ requires a transversal projectile polarization in the scattering plane

(P = Pj,x) and also a longitudinal component (P = P^i)-

Finally, it should be noted that in the presence of explicitly spin dependent forces such as the spin-orbit interaction, a "complete experiment" can only be performed for Jq = 0 ^

Jj = 0 transitions, if nothing but the projectile polarization

is prepared before the collision and measured thereafter. In

this special case, the number of independent parameters is reduced to four — sufficient to determine the magnitudes

of and the relative phase between the "direct" amplitude /

and the "spin-flip" amplitude g [4]. In all cases involving

non-vanishing target angular momenta, however, there are at

least six independent complex scattering amplitudes. Hence,

the magnitudes and relative phases cannot be determined

uniquely in the above arrangement. On the other hand,

the neglect of explicitly spin dependent forces (for example,

in the approximation of the pure "fine structure effect" [5]) will reduce the number of independent scattering amplitudes

and may lead to symmetry relations between the above

generalized STU-parameters [1]. The experimental and

theoretical test of such relationships can then provide a

detailed test about the importance of these spin dependent

forces.

1 t I r r

Figure 1: Physical meaning of the generalized STUparameters: apart from the differential cross section cr„ for the scattering of unpolarized projectiles from unpolarized

targets, we show the polarization function 5p, the asymmetry

function 5^, the contraction parameters T^, T^, and

finally the parameters U^^ and U^^ that determine the

rotation of a polarization component in the scattering plane.

2.2 Electron-Photon Coincidences and Integrated Stokes

Parameters

Another important case are the electron-photon coincidence

experiments, where (up to now) the final projectile polarization has not been observed directly (in some special cases, however, it can be calculated from experimental data through

certain linear combinations of scattering amplitudes [6]). Hence, reduced density matrix elements p^,j^^ {0) can be

constructed from eqn. (3) by summing incoherently over the unobserved final spins {m\ = mj. These matrix elements

describe the target state after the collision, but they simultaneously depend on the projectile scattering angle 0. They

can therefore be determined either in coincidence experiments

or via laser-assisted superelastic scattering processes which

provide the equivalent information [7]. However, if the scattered projectiles are not observed

at all, we can construct "integrated" reduced density matrix

elements Pj^/'m integrating over all scattering angles.
66

In this report, we will concentrate on these elements which

can be experimentally determined by measuring the so- called "integrated" Stokes' parameters. The cases of highest

practical interest are the following:

i) Transversely polarized electrons (P = ^y^y) incident

along the z-axis and observation of the light in the

y-direction defined by the incident electron polarization. In this case, we can observe the following integrated

Stokes' parameters:

the degree of linear polarization with respect to the z and X axes

^3 = ^'l =

7(0°) - 7(90°)

7(0") + 7(90°)

(5a)

where 7(/3) denotes the intensity transmitted by a Nicol

prism oriented at an angle /3 with respect to the z-axis; the linear polarization

^1 7(45°) -7(135°)

' 7(45°) + 7(135°)

and the circular Ught polarization

(5b)

(5c)

where and 7_ are the intensities of hght transmitted

by polarization filters which only admit photons with

positive (7_,.) and negative (7_) helicity, respectively. In eqns. (5), the 77's correspond to the notation used,

for example, by Blum [2] while the P's are the light

polarizations defined by Born and Wolf [8]. Longitudinally polarized electrons (P = P^e^) and

observation of the Ught along the z-direction. For symmetry reasons, only the circular light polarization

7?2 = — -P3 can be different from zero in this case. In our earlier work [9], it has been shown that for = 0''^^" Jj = 1°'^'' transitions the circular light

polarization V2/Py *s always positive, provided explicitly

spin dependent effects can be neglected completely. It was therefore not very surprising to find large deviations

from these predictions for electron-mercury scattering

[10-12], in particular because of the "intermediate

coupUng" character of the atomic wavefunctions. A

more meaningful statement about the importance of spin dependent effects on the continuum electron can

therefore be made for electron-alkaU scattering which

has been examined experimentally in the Mainz and

Munster groups [13-15]. In fact, we have recently

shown [16] that in the "LS-approximation" (only spinflips through exchange processes), the various integrated

Stokes' parameters are not independent of each other. This allows not only a qualitative , but also a quantitative

test of the importance of explicitly spin dependent

effects, as wiU be shown below.

2.3 Electron Impact Ionization with Polarized Particles

Our last example considers experiments where polarized

atoms are ionized (or excited) by polarized electrons, but

only the total ionization (or excitation) cross sections are measured. As before, the preparation density matrix of the

colhsion partners is described by eqn. (2). From eqn. (3), we can then construct a reduced density matrix by summing

over m'l = rrij, integrating over all possible directions of kj

and, for excitation, by also summing over M{ = Afj. For

ionization processes, however, the latter summation has to be

split into a sum over = Mp i.e., the magnetic sublevels

of the residual ion, and a second integration over all electron

ejection angles. In explicit numerical calculations, these steps must be

performed consecutively. It is, however, possible to make

some general statements about the overall structure of the

final result. If we denote the cross section for parallel

projectile and target polarizations [P^ and P^) by and

for antiparallel polarizations by Q^^, the cross section can be

written as [17]

QtT =Qu- PePAQpol

Qn=Qu + PePAQpol

(6a)

(6b)

where Q„ is the contribution originating from unpolarized

colhsion partners while Qp^i denotes the part depending on

the spin polarization. A spin asymmetry can then be defined

by

1 Qn-Qu _ Qpoi A = PePAQn + Qv

(7)

which allows the determination of the individual contributions to the total cross section. The usefulness of this parameter in testing theoretical models can be seen as follows. In many numerical treatments,

the ionization cross sections are written as [18]

Qu = Qd + Qe- aQ^nt (8)

thus containing a direct (Q^), an exchange (Q^) and an

"interference" contribution (o(5,„t). It can also be shown [17] that the cross section Q^^i introduced in eqns. (6) is identical

to the interference cross section a;Q,„,. Various methods have

been used to predict the above contributions via numerical

calculations. In particular. Younger [18] successfully apphed

the so-called "maximum interference approximation" with a ^ 1. This approximation minimizes the total cross section while it maximizes the asymmetry parameter in eqn. (7). Since only the interference contribution appears in the numerator, the above asymmetry is expected to be more

sensitive to the exchange part than the total cross section. This is another example of how the use of polarized collision partners can allow a more detailed test of theoretical models.
67

3. Example Results

3.1 Generalized STU-Parameters for Electron Impact

Excitation

0.00

-0.25

-0.50

'

1

' 1 Cl

• \ /: : ; / ., •VYV/'

, 1,1.

60 120 180 180

e [degl

Figure 2: DWBA results [19] for the differential cross section and the generaUzed STU-parameters for excitation

of the [5p^6s]^P2-state in xenon for an incident electron energy of 40 eV. a) absolute differential cross section a;

b) polarization function Sp ( ) and asymmetry function

5^ ( ); c) contraction functions T (- -),T, (----)

and (-

u.. (—--)•

•); d) rotation functions U^^ (- and

In fig. 2, we present the differential cross section and the

generalized STU-parameters for electron impact excitation of the [5p^6s]^P2-state in xenon for an incident energy of 40 eV. There are no experimental data for comparison, but the

theoretical curves obtained using the DWBA approach [19] exhibit the following interesting symmetry properties:

-5.

0.4

0.6

(9a)

(9b)

(9c)

(9d)

It is indeed possible to derive these relationships by using

the approximations of the pure fine-structure effect where

the energy splitting of the multiplet as well as the spin-orbit

interaction between the continuum electron and the target

nucleus are neglected and spin-dependent effects are only

taken into account by an algebraic recoupling procedure [5]. This approximation may be roughly valid for the above case; however, experimental tests of the symmetry predictions as

well as the actual theoretical values are highly desirable.

3.2 Integrated Stokes' Parameters for Electron

Scattering

Alkali

As mentioned above, is can be shown that in the absence

of exphcitly spin dependent forces (LS-approximation), the

integrated Stokes' parameters introduced in eqns. (5) are not independent of each other [16]. Hence, a quantitative test of this approximation can be performed by comparing the predicted correlations either directly with experimental data

or by using theoretical results obtained with the inclusion

of the above effects. We have tested the predictions of the LS-approximation against the results of the R-matrix (close- coupling) calculation by Scott et al [20]. This calculation

explicitly included the most important relativistic effect, namely the one-electron spin-orbit term of the Breit-PauU

hamiltonian (Bethe and Salpeter [21]). We can therefore expect at least some disagreement between the predictions

of the LS-approximation and the numerical results. The

magnitude of these deviations should give an indication about

the importance of the spin-orbit interaction on the excitation

process.

Table 1

Stokes' peirameters for e - Cs scattering

energy [ eV] "exact" calculated

1.63 9.044

ril{\)IPy[%] 1.63 0.121

%] 1.63 17.040

m)iPv[%] 1.63 -12.083

^2il)IPA%] 1.63 14.738 13.288 m)IPA%] 1.63 8.120 6.521 mm 2.04 8.049 VUDlPyi%] 2.04 - 0.122 VUDlPyi%] 2.04 18.014 m)iPv[%] 2.04 -11.985

m)IPA%] 2.04 14.340 14.623 m)iPA%] 2.04 6.984 5.123

Table 1 summarises the results obtained for the

Stokes' parameters of the hght emitted in the transitions (6p)2p°/2 -» (6s)2S^/2 and (6p)2p°/2 -> (6s)2S^/2 ^^^^ium

after excitation by polarized electrons [16]; the superscript in the 7/'s indicates the direction of observation, and the total angular momentum Jj of the excited target state is given in brackets. The column labelled "exact" corresponds to the

results obtained directly from the K-matrices of the above cal- culation (also see Nagy et al [22]) while the column labelled
68

"calculated" contains the results for V2(2y^z ^2(|)/-^j

obtained by determining the necessary three independent

variables from the other Stokes' parameters and using them

to calculate these light polarizations. It can be seen that the

deviations between the "exact" and the "calculated" values

are quite significant in some cases (up to almost 30%). This

is most likely due to the very complicated resonance structure

in the above near-threshold energy region. In any case, these numbers illustrate that the widely used conceptions of

"singlet" and "triplet" or "direct" and "exchange" scattering

are only vahd to a limited extent for this colUsion system.

3.3 Ionization Asymmetries

30 40 SO Energy (eV)

Figure 3: Spin asymmetry A for electron impact ionization

as a function of the incident electron energy. The theoretical results were obtained in a DWBA approach combined with the "maximum interference approximation" [23]. The individual curves for the initial states [ls2s]^S^ of helium ( ), [ls22s]2S^ of lithium ( ), [2p^3s]2S^ of sodium ( •) and [3p^4s]2S^ of potassium

( ) are compared with the experimental data of the Bielefeld group (U, lithium [24]; sodium [24]; A,

potassium [24]; o, heUum [25]).

In fig. 3, we show results for the ionization asymmetry A

defined in eqn. (7) for the ground states of the alkah

atoms lithium, sodium and potassium as well as for the

excited [ls2s^S^] state of helium. The theoretical values were obtained using the DWBA-method in connection with

the "maximum interference approximation" [23]. While

the agreement with the experimental data of the Bielefeld group [24,25] is quite satisfactory at higher collision energies,

there are significant discrepancies between theory and exper- iment near the ionization threshold. Since the theoretical values lie below the experimental data in tbJs region, this

result indicates a fundamental problem with the DWBAapproach (recall that the "maximum interference approximation" already maximizes the theoretical asymmetry in this

calculation). Hence, the more detailed experiment involving

polarized collision partners revealed again some defects in a theoretical approach which is rather successful in the

description of the total cross section for unpolarized particles.

Finally, we note that the theoretical results look very

similar for all four targets investigated in this study while

the experimental data for potassium differ significantly from

those for the other systems. Some possible explanations

for this feature have been proposed, however, more work is required before a definite conclusion can be drawn.

4. Summary ., ^ i

In this report, we have shown how polarized collision partners can be used for detailed testing of theoretical models. A

quantitative test for the importance of explicitly spin dependent forces has been proposed. Finally, it was shown that the

"maximum interference approximation" which has been used

very successfully to predict total ionization cross sections,

exhibits significant defects when applied to the calculation of spin dependent ionization asynmietries.

Acknowledgments

I would like to thank Professors G. Baum, K. Blum, M. Fink,

G.F. Hanne, J. Kessler, W. Raith and H. Steidl for stimulating discussions on various topics. Financial support from

the Deutsche Forschungsgemeinschaft in Sonderforschungsbereich 216, the NATO under grant #86/353 and the

Research Corporation under grant #C-2640 is gratefully

acknowledged.

References

[1] K. Bartschat, Physics Reports (1989), in press

[2] K. Blum: Density Matrix Theory and Applications (Plenum, New York 1981)

[3] O. Berger and J. Kessler, J. Phys. B 19 (1986), 3539

[4] J. Kessler: Polarized Electrons, 2nd edition (Springer,

Berlin 1985)

[5] G.F. Hanne, Phys. Rep. 95 (1983), 95

[6] J. Goeke, G.F. Hanne and J. Kessler, Phys. Rev. Lett. 61 (1988), 58

[7] I.V. Hertel, M.H. KeUey and J.J. McClelland, Z. Phys.

D 6 (1987), 163

[8] Born and Wolf: Principles of Optics, 4th edition

(Pergamon, New York 1970)

[9] K. Bartschat and K. Blum, Z. Phys. A 304 (1982), 85

[10] A. Wolcke, K. Bartschat, K. Blum, G.F. Hanne and

J. Kessler, J. Phys. B 16 (1983), 639

[11] K. Bartschat, N.S. Scott, K. Blum and P.G. Burke,

J. Phys. B 17 (1984), 269

[12] K. Bartschat and P.G. Burke, J. Phys. B 19 (1986),

1231

[13] N. Ludwig, A. Bauch, P. Nass, E. Reichert and

W. Welker (1986), Z. Phys. D 4, 177
69

[14] P. Nass, M. Eller, N. Ludwig, E. Reichert and M. Webersinke (1989), Z. Phys. D 11, 71

[15] F. Eschen, G.F. Hanne, K. Jost and J. Kessler (1989),

J. Phys. B 22 (1989), in press

[16] K. Bartschat, J. Phys. B 22 (1989), in press

[17] G. Baum, E. Kisker, W. Raith, W. Schroder, U. Sillmen

and D. Zenses, J. Phys. B 14 (1981), 4377

[18] S.M. Younger, Phys. Rev. A 22 (1980), 111

[19] K. Bartschat and D.H. Madison, J. Phys. B 21 (1988),

2621

[20] N.S. Scott, K. Bartschat, P.O. Burke, O. Nagy and

W.B. Eissner, J. Phys. B 17 (1984), 3775

[21] H.A. Bethe and E.E. Salpeter: Quantum Mechanics

of One- and Two-Electron Atoms (Springer, BerUn

1957)

[22] O. Nagy, K. Bartschat, K. Blum, P.O. Burke and

N.S. Scott, J. Phys. B 17 (1984), L527

[23] K. Bartschat, J. Phys. B 22 (1989), submitted

[24] G. Baum, M. Moede, W. Raith and W. Schroder,

J. Phys. B 18 (1985), 531

[25] J. Taborski, G. Baum, M. Fink, W. Raith and H. Steidl

in: Verhandlungen der Deutschen Physikalischen

Gesellschaft (Spring-Meeting Borm 1988)
70

ALIGNMENT AND ORIENTATION IN SIMPLY-STRUCTURED

ATOMIC COLLISION COMPLEXES

Rainer Hippler

Fakultat fiir Physik, Universitat Bielefeld, D-4800 Bielefeld 1, Fed. Rep. Germany

Electronic processes in simply-structured atomic

collision complexes are of fundamental importance

for our understanding of collision dynamics and also

of practical relevance for a variety of experimental

fields including fusion technology and astrophysics.

In continuation of our previous work^~~^ we are currently investigating alignment and orientation in

such collision systems. We concentrate our investi- gations on the excitation of atomic hydrogen to the

H(2p) state either by direct or by charge changing

processes. The quantities of interest here are the

so-called alignment A20 and the orientation Oi.^ The

alignment A20 provides detailed information about

the relative population of the H(2p„i) magnetic substates; it is defined as''

120 a{2p) (1)

where ctq and ai (=cr_i) are the total cross sections

for excitation into the H(2po) and H(2pi) magnetic

substates, respectively, and o'(2p) = ctq -f 2 ai.

Experimentally, A20 is obtained from a measurement

of the linear polarization Pi of Lyman-Q radiation

emitted during the decay of the excited H(2p) state

to the H(ls) ground state. Here we define the degree

of linear polarization Pi as

Pi h - h

(2)

and where /y and /j. are the intensities of Lyman-a

radiation with electric vector parallel and perpendicular, respectively, to the incident projectile direction

(z-direction). The orientation Oi is a measure of the

transferred angular momentum < > in a direction

perpendicular to the scattering plane,

Ox = -I: < Ly >, (3)

and where the scattering plane (x-z plane) is de- fined by the directions of the incident and scattered

projectile. The transferred angular momentum in a given direction is directly related to the circular

polarisation P3,

n-)-H+)

(4)

in the same direction. /( — ) and I {+) denote the

intensity of circularly polarized light with negative

and positive helicity, respectively.

For the simplest one-electron system, H(2p)

excitation in

H+ + Hi2p)

H{2p) + H+

(5)

collisions is predicted to proceed at low incident ve- locities V through a united-atom 2p(T~2p7r rotational

coupling in the transiently formed quasi- molecule

(Fig. 1). This should lead to population of the

H(2p±i) magnetic substates only.

Fig. 1: Schematic energy-level di- agram for H+-H. The binding en- ergy of an electron moving in the

field of two approaching protons is shown as a function of the internuclear separation R. For clarity, the

/-degeneracy has been removed.

As is indicated in Fig. 2, the measured inte- gral alignment^ is rather large at small projectile

energies and close to the majcimum value of -f- 50%

permitted by Eq. (1). These measurements are thus in good agrement with theoretical calculations

which properly account for this rotational coupling

mechanism.^" Other calculations, for example
71

coupled-state calculations employing separated-atom

electronic wavefunctions only/^ are at lower velocities at variance with our experimental results. A different picture emerges for the quasi- one

electron system

H + He-^ H{2p) + He (6)

which was investigated in an incident energy range

1-25 keV. Fig. 3 displays the measured integral

alignment for this collisions system, which differs significantly from our results for H"^ ~ H collisions. Again, as becomes obvious from a corresponding

correlation diagram,^''^ a united-atom 2p(T-2p7r rotational coupling provides an important mechanism for H(2p) excitation in H-He collisions at low incident

velocities. In addition, radial couplings occur among

the 2pa" and the 2sa and 3dcr orbitals, resulting in a negative A20 for small incident velocities. Differ- ential measurements show that the radial couplings

are most pronounced for small impact parameters,

whereas the 2p(7-2p7r rotational coupling dominates

at larger impact parameters (Fig. 4).

afi>

20 ro

-10

- 10

3

TO

Ol

—J"Do

20 05 1 20 30 —

Energy ( keV

Fig. 2: Integral ahgnment A20 (left- hand scale) and degree of linear

polarization Pi (right-hand scale)

for H(2p) production in (•) 11+ - H collisions versus incident energy.^

Also shown are the predictions of

various theoretical calulations (see

text).

Towards larger velocities the integral alignment

becomes positive. This result is an indication of a direct ls-2p excitation process which at medium

velocities favours the H(2p-i-i) magnetic substates;

it is, however, only in qualitative agreement with

optical potential caJculations of Ast et al.^'* and

with model calculations according to Andersen and

Nielsen.

20-

<=> 0 <

-20

-30 -

T 1 1 1 1—TTTT

UNI8I 1 1989-Z

1 r

H +He —H(2p) + He

Hi

5 10 Energy ( keV

Fig. 3: Integral alignment A20

for H(2p) production in (0) H-He

collisions versus incident energy. Also shown are theoretical calculations of Kimura and Lane^^ (dashed

line), optical potential calculations

of Ast et al.^'* (dash-dotted line),

and model calculations according to Andersen and Nielsen^^ (solid line]

T

^20

-0.5 -

-1.0 -

Impact parameter (au)

Fig. 4: Differential alignment A20

versus impact parameter 6 for D(2p)

production in (A) 2 keV and (o) 3 keV D-He collisions. Also shown

are corresponding calculations of

Kimura and Lane^^ (dashed line,

soHd line).
72

The measured orientation was only weakly de- pending on the impact parameter. It was observed,

however, to display a rather pronounced energy de- pendence (Fig. 5). This behaviour is explained by

the development of the adiabatic phase

X. h - I E,{R)dR, (7) V J

which is different for Xi and Xq due to the differ- ent binding energies (R) of the quasi-molecular

states along which the complex excitation amplitudes /i and /o evolve.^ As the collision velocity V increases, the phase difference Xi — Xq decreases

with increasing projectile energy; since the orientation is proportional to sin(Xi— Xq) a pronounced

velocity and, hence, energy dependence of < > is observed.

Impact parameter (au)

Fig. 5: Transferred angular mo- mentum < Ly > versus impact pa- rameter b for (A) 2 keV and (o) 3

keV D-He collisions. Dashed and

solid curves are corresponding cal- culations of Kimura and Lane.^^

The experimental results for the neutral

H (Is) + H (Is) ^ H (2p) + H (nl) (8)

collision system (Fig. 6) indicate that H(2po) and

H(2p±i) states are populated with about equal probability below E < 10 keV. This results is at variance with recent theoretical calculations,^* where

the electron wavefunction has been expanded in

travelling atomic orbitals on both centers. Molecular orbitals are not included in these calculations

and, hence, the above mentioned 2pa-2pn united

atom rotational coupling mechanism is not properly accounted for. In fact, when comparing these

calculations for H-H with previous atomic orbital

calculations for H+-H collisions (e.g., Fig. 2), we

note that even on a quantitative basis they predict a rather similar energy dependence of ^20- This

points to the necessity of including united-atom

wavefunctions in the calculations. The inclusion of

united-atom couplings is more complicated in H-H

collisions, however, where above E > I keV both

the ionization and the charge exchange (H -f H —> H+ + H") channel become comparable or even

larger than the H(2p) excitation channel. Recent calculations of Borondo et al.^^ have shown that

population of H(2s) and H(2p) states in H-H col- lisions is in fact influenced by a long-range radial

coupling with the —^H"*" + H" charge exchange exit

channel.

-40-

2 5 10 Energy ( keV )

20

Fig. 6: Integral alignment .420 for H(2p) production in (o) H+-H and

(•) H-H collisions versus incident

energy. Also shown are theoretical

calulations of Singal et al.^^ for H-H

(solid line) collisions.

Two-electron systems, for example H-H or (HHe)+, are of considerable interest as they allow an

investigation of spin and, hence, of electron-electron

interaction effects during the collision. For the (HHe)"*" system we have investigated the following two

reaction channels

H{2p) + He-^ (9)

+ He

for which, in an independent-electron picture and at low velocities, we could expect the same A2o. As is
73

illustrated in Fig. 7, excitation to the H(2p) state

in He''"-H collisions occurs dominantly at small in- ternuclear separations via couplings of the 2p(T MO

with other near-degenerate MOs (for example, 2p7r or 2scr) In H^-He collisions, on the other hand,

H(2p) excitation is thought to arise from a two-step

mechanism. In a first step a long-range radial coupling populates the charge exchange channel He"*"-!!; then the second step, as before, involves couplings of

the 2pcr MO with other high-lying MOs leading to H(2p) excitation. Thus, although the two incident

channels are different they appear to proceed along

the same intermediate states and through the same

couplings; in a one-electron picture, these states and

couplings are identical for H"'"-He and He"*"-!! colli- sions. If this picture were correct, the similarity of

H'^-He and He"^-!! would be reflected in certain characteristic signatures of the collision. For example,

the alignment depends sensitively on the contributions of a and tt states to the excitation process. In a one-electron picture there is little reason why the

two collision systems would produce a different alignment. In a two-electron picture, on the other hand,

the difierent spin channels have to be considered: In

H'''-He collisions there exists only one (singlet) spin

channel, whereas in He"*"-!! collisions both singlet

and triplet channels can contribute. Therefore, we

expect a different behaviour of the two different inci- dent channels of the (H-He)''" collision complex, and

in turn a different alignment for H'^-He and He"*"-!!. H*+He (3s)

R-'+HelZp)

H(n = 2) + He*

H*+He(2s)

As we have expected, a marked difference betv/een the two different incident channels of the

(H-He)"*" collision complex is observed. ^^""^'^

H*+He{1s;

40

30

20

°^ 10 oCM < 0

-10

-20

-I

~i \ I I I I I ! J"

• He*-H

o - He

Fig. 7: Schematic ( diabatic ) energy

level diagram for (H-He)"^ collisions

(singlet case).

0.5 1 2 5 10 20 Energy /mass ( keV/nucleon)

Fig. 8: Integral alignment A20 for H(2p) production in H^-He and

He"*"-!! collisions versus incident en- ergy divided by the projectile mass. Experimental results for He^-II col- lisions (•, Ref. 24) and for H'^-He

collisions (o, Ref. 22; , Ref. 23)

are compared with corresponding

theoretical calculations (dashed and

soUd lines, Ref. 24) and with recent

calculations for He"*"-!! by Errera et

al^^ (dash-dotted line).

In the following we shall discuss the origin of

this difference in more detail. This is facilitated

by the fact that the (H-He)+ collision complex has

been studied quite thoroughly. In particular, re- cent differential measurements^ have demonstrated

that for incident H+-He collisions the 2pcr-2p7r ro- tational coupling dominates at impact parameters

b ~ 1 a.u.; this excitation mechanism populates the

H(2p±i) magnetic substates (Fig. 9). At smaller b a double-rotational coupling 2pa — 2p7r — 2sa leads

to H(2po) excitation. This 2sa state population is produced by the radial coupling among the 2sc7 and

2pcr states around R ^ 0.4 a.u.. At low incident en- ergies of a few keV, the integral populations caused

by these two couplings are roughly equal; as a conse- quence, the integral alignment A20 is approximately

zero. This does not hold for the reverse system He+-

H, where we observe a significantly larger (positive)
74

alignment. It indicates that the above doublerotational coupling mechanism is now of a lesser importance. This experimental result is confirmed

by theoretical calculations, in which the molecular

states for both singlet and triplet manifolds were obtained by the full configuration interaction method.^"*

The most notable difference is that, for the triplet

manifold, the separate atom level H^-f He(2^P) lies lower than the H(n:=2)-|-He"*" levels, while the situation is just the opposite for the singlet manifold. In the singlet case, H(2p-ti) population is produced by

a 2pcr-2p7r rotational coupling, whereas H(2po) ex- citation is produced by the 2p<7-2p7r-2s(T coupling.

At low energies, these contributions produce a and

TT states with comparable strength, and the integral

ahgnment in H"'"-He collisions (pure singlet case)

is small. The same couplings also operate in the

triplet manifold; however, the important difference

is that here they do not lead to excitation of the

H(2p) states. This is because for the triplet channel

the 2p<T state, for example, does no longer correlate with the He+-fH(n=2) outgoing channel. This

leads to the results that H(2p) excitation in He^-H

collisions at low incident energies is dominated by

a rotational coupling mechanism leading to population of outgoing TT-states. It has the consequence

that predominantly the H(2p±i) magnetic substates

become populated, which is reflected in the positive

alignment.

10

0.8

to 0.6

*o

O.A

11 0.2

0.0

T 1 1 —1——

H'-He^H(2p).He*

0.2 0.6 10 14

Impact Parameter (a.u.)

Fig. 9: Relative population of

H(2po) versus impact parameter b

in H+-He collisions at incident energies of 1-4 keV. Solid line is a

theoretical calulations of Macek and

Wang;^^ the other two curves are from Fritsch^*^

As the incident energy increases, direct (impulsive) excitation to H(2p) through a-a couplings

becomes important resulting in preferred H(2po)

population in both collision systems. This causes

the sign change of A20 at « 15keV/nucleon. Above

10 keV/nucleon the theoretical curves for H''"-He and

He+-H collisions appear to approach each other; this behaviour is not unexpected since the spin effects discussed here should be of minor importance at large collision energies.

Polarization studies in ion-molecule collisions have been performed as well. For example, the

degree of linear polarization in H+-H2 collisions was found to be rather large (P ^ —12%) around incident

energies of 1-2 keV (Fig. 10).

-I—I—I—I I I 1

o H* - H2 • He*- Hi

"T 1 1—I—I I I I

I

1 r

w ^

_i I

I I I I 1

Jl I I I I I I I 1_

0 5 t 2 5 10 20 30 Energy/mass ( keV/nucleon )

Fig. 10: Degree of linear polarization versus incident energy divided

by the projectile mass for (o) H+-H2

and (•) He+-H2 collisions.^*

For this collision system the observed radiation

should result either from projectile excitation to the

H(2p) state [—> H(2p) + Hj] or from dissociative ex- citation of the target H -j- H(2p) + H+]. Hence,

assuming the emitted radiation to be due to H(2p)

production, this corresponds to a subshell cross sec- tion ratio Ui/ao w 2. This result is simJliar to

collisions where the 2pcr-2p7r rotational cou- pling dominates the excitation process. In a quasidiatomic approach one may invoke the same type

of coupling, although in the triatomic picture^^ the

two 2a'-3a' and 2a'-~la" transitions may lead to the

excitation of H(2p) atoms. Towards larger energies

the linear polarization in H"''-H2 collisions becomes

less negative, indicating the increasing importance of

(direct) excitation to the H(2po) substate.

The experimental results for He+-H2 collisions display a negative linear polarization amounting to
75

about —10% in the range of incident energies between 4-25 keV. For this collision system dissociative

target excitation to H(2p) provides the dominating process. Again, the observed magnitude of the

linear polarization may be taken as an indication

for a rotational coupling dominating the excitation

mechanism at these energies.

Acknowledgements

The author acknowledges helpful discussions with

Prof. Dr. H.O. Lutz, Prof. Dr. H. Kleinpoppen,

Dipl. Phys. H. Madeheim, Dr. M. Kimura, and Dr. D Dowek. This work was supported by the Deutsche

Forschungsgemeinschaft in SFB 216.

References

1. R. Hippler, M. Faust, R. Wolf, H. Kleinpoppen,

H.O. Lutz, Phys. Rev. A 37 , 4644 (1987).

2. R. Hippler, W. Harbich, M. Faust, H.O. Lutz, J. Phys. B 21 . 103 (1988).

3. U. Fano, J. Macek, Rev. Mod. Phys. 45, 553 (1973).

4. R. Hippler, In: Fundamental Processes in Atomic

Collision Physics (H. Kleinpoppen, J.S. Briggs,

H.O. Lutz, Eds.), Plenum: New York (1985), p. 181; R. Hippler, In: XV ICPEAC Electronic and

Atomic Collisions, Invited Papers (H.B. GUbody,

W.R. Newell, F.H. Read, A.C.H. Smith, Eds.),

North-Holland: Amsterdam (1988), p. 241.

5. R. Hippler, H. Madeheim, W. Harbich, H. Kleinpoppen, H.O. Lutz, Phys. Rev. A 38, 1662 (1988).

6. M. Kimura, W.R. Thorson, Phys. Rev. A 24 , 1780

(1981) .

7. W. Fritsch, CD. Lin, Phys. Rev. A 26 , 762 (1982);

Phys. Rev. A 27 , 3361 (1983).

8. H.J. Liidde, R.M. Dreizler, J. Phys. B 15 , 2703

(1982) .

9. T.G. Winter, CD. Lin, Phys. Rev. A 29 . 567 (1984).

10. R. Shingal, CD. Lin, private communication (1989).

11. D. Rapp, D. Dinwiddle, J. Chem. Phys. 57, 4919

(1972).

12. C. Gaussorgues, A. Salin, J. Phys. B_4, 503 (1971).

13. R. Shakeshaft, Phys. Rev. A 18 , 1930 (1978).

14. J. Ast, H.J. Liidde, R. Dreizler, J. Phys. B 21, 4143

(1988).

15. N. Andersen, S.E. Nielsen, Adv. At. Mol. Phys. 18, 265 (1982).

16. H. Madeheim, R. Hippler, unpublished.

17. M. Kimura, N.F. Lane, Phys. Rev.A 37 , 2900 (1988).

18. R. Shingal, B. H. Bransden, D. R. Flower. J. Phys.

B 22 , 855 (1989).

19. G.W. McClure, Phys. Rev. 166, 22 (1968).

20. M.W. Gealy, B. Van Zyl, Phys. Rev. A 36 , 3100

(1987).

21. F. Borondo, F. Martin, M. Yanez, Phys. Rev. A 35 , 60 (1987); Phys. Rev. A 36 , 3630 (1987).

22. R. Hippler, W. Harbich, M. Faust, H.O. Lutz,

L. Dube, J. Phys. B 19 , 1507 (1986).

23. P.J.O. Teubner, W.E. Kaupilla, W.L. Fite, R.J. Girnius, Phys. Rev. A^, 1763 (1970).

24. R. Hippler, H. Madeheim, H.O. Lutz, M. Kimura,

N.F. Lane, Phys. Rev. A (1989), to be published.

25. L. F. Errera, L. Mendez, A. Riera, Z. Physik D, to be published; L. Mendez, private communication.

26. J. Macek, C. Wang, Phys. Rev. A 34 . 1787 (1986).

27. W. Fritsch, private communication.

28. D. Dowek, H. Madeheim, R. Hippler, XVI ICPEAC,

Book of Abstracts (1989), and to be published.

29. V. Sidis, D. Dowek, XIII ICPEAC Electronic

and Atomic Collisions, Invited Papers (J. Eichler, I.V. Hertel, N. Stolterfoht, Eds.), North-Holland:

Amsterdam (1984), p. 403.
76

IS THE "CLASSICAL" DOUBLE SCATTERING LESS ACCURATE FOR CALIBRATING MOTT

POLARIMETERS THAN RECENTLY PROPOSED METHODS?

K. Jost ^

Physikalisches Institut, Universitat, D-4400 Miinster, FRO

In most experiments involving spin polarized

electron beams, the degree of (transverse) polarization

P of the beam is determined by a Mott analyzer.^ In

this instrument the polarized beam is scattered by a

target foil of high atomic number (usually gold, Z = 79) at an energy E in the 100 keV region; in modern analyzers using a retarding field for energy

analysis the energy is sometimes as low as 20 keV or even 150eV.2.3 Due to spin orbit interaction, the

scattered intensities L and R observed simultaneously

by two counters under the same angle 6 on the left and on the right in a plane perpendicular to P show

cin asymmetry

A = (L-R)/(L+R) = SP, (1) which is proportional to P. For the constant S in this relation exist a variety of names, among these the most

popular ones seem to be "analyzing power" or "Sher- man function". In theoretical calculations, S is a function of Z, 6 and E. However, because in a real experiment multiple scattering and other disturbing effects cannot be avoided, from the experimenter's viewpoint

S depends additionally on many other parameters such

as foil thickness t, solid angle Afi accepted by each of

the two counters and their energy resolution AE. In

principle the inclination of the target foil with respect

to the incoming beam is of importance as well.^ However, in order to avoid drastic instrumental cisymmetries, only the "symmetric" arrangement (normal inci- dence) is popular for Mott analyzers (Fig. 1).

Fig. 1. Mott analyzer with symmetric foil position

As the theoreticcil value of S is only valid for

elastic single scattering, it is important to distinguish

between the effective analyzing power Sgff of a real analyzer and the idealized one. If Seff of a Mott analyzer is known, it follows immediately from Eq. (1) that aa unknown polarization can be determined by meeisuring the asymmetry:

P = A/Seff. (2)

^ deceJised

In the following, an old and several recently pro- posed methods for determining Seff will be discussed.

At first the methods will be described neglecting dis- turbing effects due to imperfections of the apparatus or of the beam alignment. However, some of these methods are plagued heavily by false asymmetries and it is the main purpose of this contribution to show a strat- egy on how false asymmetries can be controlled.

L CALIBRATION METHODS

A. Double scattering

The oldest^ and the most straightforward method

to determine Seff absolutely is the double scattering of

electrons. This method takes cidvantage of the theoretically well established equality of the polarizing and

analyzing power (see, e.g., Ref. 1, section 3.4.5 or Ref.

6, Fig. 15): Putting another foil in front of the Mott

ajialyzer and scattering an unpolarized electron beam

from it (Fig. 2), the polarization of the scattered beam

will be Seff) provided that all relevant parameters (Z,

6, E, t, normal incidence...) are the same as in the

amalyzer, where the "second" scattering occurs (nota

bene: multiple scattering occurs in both foils in like manner). Therefore, according to Eq.(2), one should

obtain S^eff by measuring A. However, experience with

this method shows that it is not as simple to handle as described above because of the difficulty to control spurious asymmetries (see sections II and III).

Fig. 2. Double scattering experiment

Therefore, this method is usually only applied

indirectly by omitting the first scattering process. Instead, a set of foils of different thicknesses and a

source of polarized electrons with unknown (but stable)

polarization is used. Extrapolating the measured

asynrmietries to zero foil thickness and identifying the
77

analyzing power in this case with the idealized one

yields the formerly unknown P of the source and in turn the value Seff of the foil actually used in the

analyzer. However, this procedure has several pitfalls:

(i) The extrapolation procedure is not unique^ because

there exists no reliable theoretical guide on the dependence of Seff on the foil thickness t. A much more

extensive Monte Carlo calculation than the only existing one^ would be an invaluable help to cope with

this problem, (ii) The ideal value of S is not unique

either: For E = 120 keV and 6 = 120° , for example,

theoretical values^ "^2 range from S = —0.400 to —0.407, and the most accurate value obteiined in a real double scattering experiment —O.ST&t 0.008; i.e. between the extreme values is a difference of about

8%. (iii) An attempt to reduce the uncertainty of the

first point by using extremely thin gold foils is dangerous as well: Evaporated gold tends to condense as

islets which form no layer of homogeneous thickness

below ~40 flg/cm"^. Therefore, apart from the irksome

increase of the contribution of the backing, foils of (macroscopically) low areal density may behave hke

thicker ones. TaJcing all three problems into consideration, the overall uncertainty obtmned by the extrapolation method could be even larger than the ±5% esti- mated by Fletcher et al.^

Because this situation is certainly not satisfactory, several alternative methods for determining Seff with

higher accuracy have been devised and explored recently. All these methods are ba^ed on a source of

polarized electrons with eeisily reversable polarization

and an auxihary target between this source eind the

analyzer which is to be calibrated. These new methods

eire described in the following.

B. Optical method

In this method an auxiliary target is used, which

emits circularly polarized Ught when it is excited with

polarized electrons. In the special ccise of the (unresolved) 3^P 2^Si transition in helium, the circular polarization of the corresponding 388.9 nm line is given

by (Eqs. 10 and 11 of Ref. 15):

712 = (0.5 - 7y3/6)P. (3a)

The value of 7)2 is determined by measuring the intensities of right-handed [l((T*)] and left-handed [l((J )] circularly polarized light in y direction (Fig. 3):

ri2 = [l{a') - I{a-)]/[l{a*) + lia')]. (3b)

The small correction 7/3/6 is due to an alignment along

the z axis (Fig. 3) which the excited atomic state obtains by the collision. The value of 7/3 can be determined by measuring the intensities 1(0" ) and 1(90° ) of

lineeirly polarized light emitted in y direction:

7?3 = [1(0°)- 1(90° )]/[l(0°) + 1(90°)]. (3c)

Fig. 3. Experimental arrangement for optical

, calibration of Seff with a helium target

Eq. (3a) should be valid to high accuracy, because of the low atomic number of helium and because of its zero nuclear spin; i.e. the LS-coupling scheme used in the derivation's jg valid, spin flips can be neglected

aJid hyperfme corrections are not required. However,

note that Eq. (3a) is only valid for electron energies

between 23.0 eV and 23.59 eV; below 23.0 eV the

desired 32P level is not yet reached and above

23.59 eV the 4^Si or higher states may be populated

which would yield an admixture of unknown polarization to the 388.9 nm line. The calibration of a Mott analyzer using this method is described in detail elsewhere'^; by measuring

T)2 and 773 at 23.3 eV electron impact energy the polarizatation P of the electron source is known via

Eq. (3a). The very same beam of polarized electrons is directed into a Mott analyzer, whose Seff is then

determined with an uncertainty in the 1% region by

mecisuring A in Eq. (2).

C. Elastic scattering from an auxiliary target

Between a source of polarized electrons with

reversable polarization Pq and the Mott analyzer,

whose Seff is to be determined, an auxihary target is introduced. This target T should show sufficiently distinct spin orbit effects such that its analyzing (or

polarizing) power St differs from zero — the more the

better — at the scattering angle and at the scattering

energy chosen (usually around 100 eV). A set of scat- tering experiments from this target yields sufficient (or even redundant) information to extract S^eff- Hopster and Abraham'^ used a Pt(lll) crystal surface as a target to perform the (scattering) experiments shown in Fig. 4. The target surface was oriented

like a mirror for 90° deflection. In the first of these experiments (Fig. 4a) the polarization Pq is measured

relatively (Seff is not yet known!) by deflecting the beam electrostatically into the analyzer without using

the target. The crystal was biased negatively for this purpose so that the beam was deflected without strik-
78

Fig. 4. Experimental arrcingements with an auxiliary target T to measure a) Aq, b) AT,IT

and c) A-l.li (cf. Eqs. 4a-e)

ing the target. The count rates Lq and Rq in this experiment yield the asymmetry Aq = (Lq—Rq)/

(Lo+Ro)i which can also be expressed in terms of Pq

and Seff (cf. Eq. 2): Ao = PoSeff- (4a)

(In practice, the experiment is repeated with reversed

polarization to eliminate different counter efficiencies and the like, see section II). The next two experiments (Figs. 4b and 4c) measure the change of the incoming polarization due to the scattering process by determing the asymmetries

AT=(LT-Rt)/(LT+RT) and Ai=(Li-Ri)/(Li+Ri).

The asymmetries At and Ai can also be expressed in terms of Pq, St and Seff because the change of a polarization by elastic scattering is well known. The general formula (Eq. 3.78 of Ref. 1) yields for the geometry

shown in Fig. 4

AT = PTSeff = Seff (4b)

and Ai = PiSeff = 7^°

p

Seff . (4c)

Simultaneously to At and Ai, the two experiments

(Figs. 4b and 4c) yield also the asymmetry At =

(IT-I-1)/(IT+Ii), where it ~ LT+RT and li ~ L-l+Ri.

This asymmetry is due to the analyzing power of the

target and can consequently (cf. Eq. 4a) be expressed

by

At = PoSt . (4d)

The four Eqs. (4a) — (4d) contain independent

information on Po, St and Seff. Therefore, by choosing

any three of them, the desired Seff can be expressed

by measured asymmetries. This way, one has the

choice between the following relations:

S2eff = Ao[AT(l+At) - Ao]/At ,

S2eff = Ao[Ai(l-At) + Ao]/At . ^^^^ S^eff = Ao(2ATAi+AoAt-AoAJ.)/(2Ao-AT+Ai.),

S2eff = [(AT(1+At))2 - (Ai(l-At))2]/4At .

In Ref. 17 only the first of Eqs. (4e) is used. However,

by variation of the electron energy at the target from

15 eV to 45 eV, St was varied as well and in this manner a set of independent determinations of Seff was performed; the uncertainty is estimated to be

within ±2%. By using a mercury atomic beam instead

of the solid Pt target, the same type of calibration of a Mott analyzer is described elsewhere (please note

that in Ref. 18 the equation which is equivalent to the

la.st one of Eqs. (4e) of the present paper contains a

misprint: the last plus sign should be a minus sign).

Garcia- Resales et aU^ used a xenon atomic beam

as an auxiliary target. They proposed a somewhat

different set of scattering experiments which are shown

in Fig. 5. In the first one (Fig. 5a), Aq = PoSeff (Eq.

Fig. 5. Experimental arrangements with an auxiliary target T to measure a) Ao,At and

b) A (cf. Eqs. 5a and b)

4a) and At = PoSt (Eq. 4d) are measured simultaneously. In this case, the intensities Ii and Ip (which

correspond to It and li in Fig. 4) are not measured in the Mott analyzer in two different runs, but simultaneously by two spectrometers equipped with electron multipliers (not shown in Fig. 5a; the spectrometers

are set to the elastic channel). The second experiment

uses an approved method20 to measure St with an

unpolarized electron beam from a conventional electron gun. To be more accurate — Seff is assumed to be

unknown — this experiment yields the asymmetry

A = StSeff . (5a)

Combining Eqs. (4a), (4d) and (5a) one obtains

SVf = ^. (5b)

At the moment, we will not evaluate the advantages or disadvantages of the different methods presented

so far. This will be postponed until instrumental asymmetries are discussed in the next section.

n. INSTRUMENTAL ASYMMETRIES

A. Causes for Lnstrumental asymmetries

Several situations leading to instrumental asymmetries are discussed in detail elsewhere (Ref. 1, sect- ion 8.1.2). Most people — including myself — feel bored

by this topic, but exactly this seems to be the reason
79

that really accurate double scattering experiments

could not be performed in the past because the long

line of (simple) arguments had been interrupted too

early. Therefore it is necessary to repeat the discussions

given in Ref. 1 and to go even a step further (section

III). In short: instrumental asymmetries are caused by

(i) different counter efficiencies (including an unintentional difference of the solid angles accepted by the

detectors or an unintentional difference of discriminator settings and the like) and/or (ii) misalignment of

the incident beam (parallel shift or oblique incidence

with respect to the ideal axis).

In order to facilitate the discussion, we will assume thin pencil beams, only small angular deviations (sinA^= A^, etc.) and positional deviations

which are small compared to counter distances from

the foil center; in the following figures these deviations

are very largely exaggerated. Furthermore, we will assume that the counters are at an angular position

where the analyzing power is not changed by a small

change of scattering angle (5Seff/5^ =0). Finally, we

will write S instead of Seff for convenience. With these assumptions, it will turn out in the following that the

first cause of instrumental asymmetry is harmless

compared to the second one, therefore we start right

away with the more difficult case. .-r >

For a moment, let us consider an unpolarized

electron beam. According to Fig. 6, an obliquely inci- dent beam which is inclined by 6 with respect to an

ideal axis and which hits the gold foil of the analyzer a

distance y away from the center changes the scattered

intensities detected by two ideal counters in two differ- ent ways: First, the (ideal) solid angle Q of the white

counter increases by

Ah

h = 2 ^ sin d , h

(6)

and second the (ideal) scattering angle d for the white

counter is increased by (Ref. 1, Eq. 8.10):

which of course changes the detected intensity I which

depends on 9. As the effects for the black counter have

just the opposite sign, one detects a false asymmetry

(even for an unpolarized electron beam!):

An

+ 1 w AO

= 2fsin^+Alr -fcos^+e

where Air = ^

} |^

(8)

is the relative change of scattered intensity with scat- tering angle. The intensity detected in the white and

black counters is proportional to (1-fAf) and (1—Af),

respectively.

Let us now return to a polarized electron beam

whose polarization may be reversable without changing

the beam position. The efficiencies of the two counters

are assumed to be different now (efficiencies W and B

for white and black, respectively). In the experiment

shown in Fig. 7a, one measures the intensities LT ~ W(l+PS)(l+Af)

and RT ~ B(l-PS)(l-Af) .

Fig. (7b) yields:

Li ~ W(l-PS)(H-Af)

and R-L ~ B(l+PS)(l-Af) .

(9)

(10)

Ad = cos 6^ (7)

Fig. 6. Oblique incidence; x, y and e are positive

in the figure

Fig. 7. Polarization reversal a) initial configuration, b) polarization reversed

Collecting the intensities with the same sign in PS and

combining them in the usual way (Ref. 1, Eq. 8.6) to define L = (LTRi)1^2 and R = (RTL-1)1^2 one obtains

A^

-- - PS L+R

- (11)

which is the ideal result; different counter efficiencies and false asymmetries are eliminated.

Unfortunately, there are also experiments in which

the incoming polarization can not be reversed. To

eliminate different counter efficiencies in this case, the
80

counter arrangement is turned by 180° around an axis which is believed to coincide with the incoming beam.

However, in general the ajcis of rotation and the beam

axis are not quite the same and one has the situation

sketched in Fig. 8. The first experiment (Fig. 8a) yields the same as Eq. (9): LT~ W(l+PS)(l+Af)

^^2)

and RT ~ B(l-PS)(l-Af).

On reversal of the counters (Fig. 8b), where the white

and black counters are interchanged, one obtains

L' T ~ B(H-PS)(H-Af)

^^3^ and R' T ~ W(l-PS)(l-Af).

Fig. 8. Counter reversal a) initial configuration,

b) counters interchanged; the dashed line shows the beam incidence which would be

equivalent to Fig. 7b

Again collecting the intensities with the same sign in PS and combining them to define L' =(LTL' T)^''^ and

R' rz(RTR' T)l/2^ one obtains (AfPS << 1 neglected,

compare Eq. 8.15 of Ref. 1):

A' =^i^|^=PS + Af , (14)

where the different counter efficiencies are eliminated,

indeed. However, the false asymmetry Af (Eq. 8) re- mains and it seems to be almost impossible to get rid of Af, e.g. by determining it with an unpolarized elec- tron beam, because this task would be successful only

when the unpolarized beam exactly coincides with the

polarized one. The encouraging result obtained by

reversal of P, Eq. (11), is not obtainable by simple means when the counters are rotated. However, it will be shown in section III how this problem can be sol- ved.

B. Influence on calibration methods

The discussion above shows that all experiments

where the polarization entering the Mott analyzer can

be reversed easily (without shifting the beam!) are

uncritical, whereas all others bear the danger of unrecognized false asymmetries. Small false asymmetries

may be tolerable in experiments with modest accuracy, but definitely not when one strives for an uncertainty

below a few percent. This may be illustrated by an

example: in an experiment with P = 0.3 and S = 0.25, one should observe an asymmetry A = PS = 0.075. Assuming a counter distance h = 30 mm (Fig. 6) and

using Rutherford scattering as a rough approximation

to determine Alp, Eq. 8 yields a false asymmetry Af

which reaches the 1% level of A = PS when the beam

has a parallel shift (c = 0) of only 0.02 mm or when it hits the center of the foil (y = 0) with an angular

deviation of only 0.04° . From discussions the author

knows that many people accept these arguments for thin pencil beams, but not for a beam with a diameter

in the region of 10 mm, where a shift of 0.02 mm

would be hardly visible. Nevertheless, a thick beam

consists of a large number of thin ones and each of these is influenced by the small shift. Summing up

yields therefore the same result for a thick beam as for a thin one. For the present purposes any thick beam

can be represented by an infinitesimally thin one (similar to the pointlike center of mass of a huge body).

In the light of the above considerations, the optical method (section LB) should be rather reliable,

because all experiments necessary to determine Seff can be performed with reversed polarization of the beam entering the Mott analyzer. All other methods

have weak points with this respect: in the classical double scattering experiment performed in the 100 keV

region any task to reverse the polarization after the

first scattering without influencing the beam is hopeless. Considerably better off is the experiment of Hopster and Abraham where the beam from all three experiments (Figs. 4a-4c) passes through a spec- trometer with narrow shts before it enters the Mott

analyzer. Therefore one can hope — if the adjustment

of the spectrometer is not touched — that the beam

enters the Mott analyzer alike in all three experiments

and Af in Eq. (14) could be measured with an unpolarized beam in the arrangement of Fig. 4a when differ- ent counter efficiencies are known also via Fig. 4a with a polarized beam (Eq. 8.8 of Ref. 1). The same type of experiments^ without narrow defining slits is very

likely disturbed by false asymmetries. Finally, the part

of the experiment of Garcia- Rosales et al^^ which is shown in Fig. 5b allows a reversal of P when the elec- tron gun is swiveled to the mirror symmetric position

in the scattering plane; however, the scattered beam

should then be defined by rather narrow slits to warrant the same beam incidence into the Mott analyzer.

It should be emphasized that the accurate calibration of a Mott analyzer is meaningless when it is intented to be used later in experiments without the possibility of polarization reversal. In this case, Eq. (14)

holds and the accurate knowledge of Seff is easily

masked by Af.
81

m. MONITOR COUNTERS

The problems with false asymmetries discussed

above can be overcome by introducing properly positioned monitor counters. The idea seems to be simple:

two additional monitor counters are placed symmetrically at small angles where S(^m) ~ 0. It is clear that

any asymmetries observed by these monitor counters

are purely instrumental. Monitor counters were popular in the past^^- 21"23^ but strangely enough, not nowadays where higher precision in electron polarimetry is of large interest. The reason is simple: everybody who

used monitor counters made the experience that the

results did not really improve and often the results obtained by observing the monitor counters seemed to be more doubtful than those neglecting them. As will be explained in the following, this strange behavior of

the monitor counters is caused by placing them at a wrong distance from the scattering center.

First of all, it should be emphasized that the

inversion of the above statement "any asymmetries

observed by these monitor counters are purely instru- mental" is not allowed. That means that ideal inci- dence is not warranted if the monitors show no asymmetry. This statement is seemingly absurd, but it can

be easily understood by looking at Eq. (8), which of course is also valid for the monitor counters if the appropriate value 6^ is inserted. Obviously, Af vanishes if both y and 6 are zero (ideal incidence), but it also vanishes if the effects of changing intensity due to a change of solid angle and scattering angle compensate each other. This is the case if

cos

2 sin^

(15)

because this condition is obtained from Eq. (8) by the requirement Af =: 0. This constant ratio e/y leading to Af = 0 for the monitor counters (indices m) is obtained for all beams which cross the ideal axis at a

distance

xm = hm/[cos^m - 2[Alr(^m)]"^ sin^n,] (16)

from the foil center, see Fig. 6. For the polarization counters (indices p) one obtains correspondingly Af =

0 for all beams which cross the ideal axis at a distance

xp = hp/[cos^p - 2[Alr(^p)]-l sin^p] (17)

from the foil center. It is an almost trivial requirement

that the polarization counters should not be allowed to be disturbed by a false asymmetry when the monitor

counters show Af — 0. On the other hand it would be

nonsense as well if the monitor counters indicate a

false asymmetry when the polarization counters are not

disturbed. These unpleasant contradictions can easily

be avoided by making the "magic points" defined by

Eqs. (16) and (17) coincide, i.e. Xm = Xp = Xo- This

Iceids to the requirement for the ratio hm/hp of the

detector distances from the foil center to be:

hn, ^ cos<?ni - 2[Alr(<?m)]"^ smO^

P 1 0r

(18) cos Obeying this requirement has the consequences that (i) any oblique beam passing through the common magic

point Xq defined above behaves with respect to asymmetries like aji ideal (axial) one and (ii) amy beam not

passing through Xq shows a false asymmetry both in the polarization counters and in the monitor counters. Moreover, it is easy to show that the false asymmetries

Afp and Afm in the polarization and monitor counters

are proportional to each other for any combination of

y and 6: Afp = [Mr{Op)/AlM] Afn, , (19)

this follows by inserting Eq. (18) into Eq. (8), which is written up for the monitor counters as well as for the

polarization counters. The use of monitor counters

positioned according to Eq. (18) at an angle 6m with

S{dm) = 0 is very simple now: because Afm registered

by the monitor counters (Eq. 14) is independent of P

and because Afp in Eq. (14) is also known via Eq.

(19), the true asymmetry PS can easily be extracted. The drastic effect of a wrong ratio of hm/hp (i.e. f Xp) is illustrated in Fig. 9: Assuming P = 0 and

using Rutherford scattering to compute Alp at 0^ =

45° and ^p = 120°, one obtains h^/hp = 1 (for Rutherford scattering independent of 0) and

A'p=0.239 A'm from Eqs. (14), (18) and (19). A plot of

A'p versus Am, Eq. (14), yields in this case a straight

line for all combinations of € between —0.2° and

+0.2° and y/hp between —0.02 and +0.02 as expected.

Using the same combinations of 6 and y/hp for the wrong distance ratio hm/hp = 5, the values of A'p and

A'm are within the parallelogram, whose corner points

are determined by the extreme combinations of € and

y/hp. From this it is clear that no unique relationship

between Afp and Afm is obtained for a wrong distance

ratio. For P ^ 0 the straight line and the parallelogram

are shifted in A'p direction by an amount PS(^p),

where S(^p) is the analyzing power of the polarization counters. , An

Fig. 9. Behavior of asymmetries A'p and A'm for a) Xm = Xp and b) Xm = 5xp. Extreme

combinations of 6 and y/hp (see text) are shown as open circles
82

Except for van Klinken^^, no other author has

recognized the importance of the ratio hm/hp (Eq. 18). However, he relied on Rutherford scattering to be a

reasonable approximation to determine Alp. Unfortunately, this assumption is completely wrong at Op =

105° which he chose. For determining hm/hp it is inevitable to measure Alp at 6p and dm- ThIS mconvenience is very profitable, however, as is shown in the

last section.

Acknowledgment: I wish to thank Professor J. Kessler

cuid especially Dipl.-Phys. A. Gellrich for numerous

fruitful discussions on the use of monitor counters. This work has been supported by the Deutsche For- schungsgemeinschaft in the Sonderforschungsbereich

216 "Polarization and Correlation in Atomic Collision Complexes".

References

rV. "CLASSICAL" DOUBLE SCATTERING

EXPERIMENT

Measurements of Seff in dependence of the foil thickness t have been made at 50 keV and 120 keV for scattering angles of 45°, 60°, 75°, 105°, 110°, 115°,

120°, 125° and 130°. The results were presented24 at the ICPEAC. The measurements (an example is shown

in Fig. 10) were highly reproducible owing to the

monitor counters used according the strategy explained

in section III. Absolute values of Seff (120°) were

determined using two equal foils of 220 flg/cm^ at 120 keV and of 70 /ig/cm^ at 50 keV. An analysis of

the uncertainties at these calibration points shows that

these are surely below 0.3%. Therefore it can be concluded that the "classical" double scattering method is by no means inferior to the recently proposed methods

for calibrating Seff if properly positioned monitor

coimters are used. Moreover, the author is convinced

that the use of monitor counters in other experiments

where the reversal of the polarization is difficult or impossible would largely improve the reliability of the measurements.

Finally, it should be emphasized again that more

extensive Monte Carlo calculations^ on the dependence

of Seff on t would allow an accurate comparison of the

ideal S (defined in the introduction) with the measurements. ^4

t(pg/cm2)200

Fig. 10 Seff versus foil thickness t obtained in a

precise double scattering experiment 24. The curves are guides to the eye only. The

size of the rectangles represent the experimental uncertainties

1 J. Kessler, "Polarized Electrons", 2nd ed. (Springer, Berlin, 1985)

2 F.B. Dunning, L.G. Gray, J.M. Ratliff, F.C. Tang, X. Zhang and G.K. Walters, Rev. Sci. Instrum. 58, 1706 (1987)

3 J. Unguris, D.T. Pierce and R.J. Celotta, Rev.

Sci. Instrum. 57, 1314 (1986) 4 H.A. Tolhoek, Rev. Mod. Phys. 28, 277 (1956),

Fig. 7

5 N.F. Mott, Proc. Roy. Soc. A 124, 425 (1929)

6 K. Jost in "Physics of Ionized Gases", M.V.

Kurepa, ed. (Beograd, 1972), p. 37

7 G.D. Fletcher, T.J. Gay and M.S. Lubell, Phys.

Rev.A 34, 911 (1986)

8 V. Hnizdo, Nucl. Instrum. Meth. 109, 503 (1973)

9 S.R. Lin, Phys. Rev. A 133, 965 (1964)

10 G. Holzwarth and H.J. Meister, Nucl. Phys. 59, 56

(1964) and Tables... (Munich 1964)

11 A.W. Ross and M. Fink, Phys. Rev. A 38, 6055

(1988)

12 W. Biihring, private communications 1987/1989

13 J. van Klinken, thesis (Groningen, 1965) and

Nucl. Phys. 75, 161 (1966)

14 L. Reimer and K. Freking, Z. Physik 184, 119 (1965)

15 T.J. Gay, J. Phys. B 16, L 553 (1983)

16 M. Uhrig, A. Beck, J. Goeke, F. Eschen, M. Sohn,

G. F. Hanne, K. Jost and J. Kessler, Rev. Sci. Instrum. 60, 872 (1989)

17 H. Hopster and D.L. Abraham, Rev. Sci. Instrum.

59, 49 (1988)

18 J. Leuker and K. Jost, XVI. ICPEAC New York

(1989), Abstracts of papers p. 803

19 C. Garcia- Rosales, H. Miiller and J. Kessler,

J. Phys. B 21, L 477 (1988)

20 K. Jost and J. Kessler, Z. Physik 195, 1 (1966)

21 A.R. Brosi, A.I. Galonsky, B.H. Ketelle and

H. B. Willard, Nucl. Phys. 33, 353 (1962) 22 W. Raith in "Atomic Physics", B. Bederson,

V.W. Cohen and F.M.J. Pichanick, eds. (Plenum

Press, New York 1969), p. 389

23 G. Schonhense, Phys. Rev. Letters 44, 640 (1980) 24 A. Gellrich, K. Jost and J. Kessler, XVI. ICPEAC

New York (1989), Abstracts of papers p. 802
83

LOW ENERGY ELECTRON SCATTERING BY 3^^3/2 SODIUM

L. Vuskovic*

Physics Department, New York University, New York, NY 10003

We have investigated partially state selective

scattering amplitudes for elastic and superelastic unpolarized electron scattering by laser ex- cited 3^P3/2 sodium using our atom-recoil scat- tering apparatus-"- . Higher level excitation measurements were also performed. Scattering in- tensity is measured with respect to polar scatter- ing angle (^), while preUminary measurements of

azimuthal (0) dependence for elastic scattering

have also been performed, although a quantitative analysis for azimuthal scattering has not yet

been made.

For the recoil technique post-collision observation is made on the recoil-scattered atoms. Absolute scattering cross sections are obtainable

from knowledge of the ratio of the scattered

to unscattered atom currents and the absolute

electron number current, while the conventional

crossed-beam method (observing the scattered

electrons), requires knowledge of the atom number density in the interaction region, a very diffi- cult quantity to determme rehably. At the same

time, the atom-recoil technique^, being suitable^

for grand-total cross section measurements, contain restrictions with respect to differential cross section observation.

Our laser excitation experiments were performed using both travelling, i.e. unidirectional,

and standing, i.e. bidirectional laser radiation.

A schematic diagram of the travelling wave ar- rangement is shown in Figure 1. The three interacting electron, atom and laser beams are mutually perpendicular, and lie along the 2, y and -x directions, respectively. The standing wave ar- rangement is also employed in the present work,

but we will not discuss it here since the corre- sponding analysis has to be modified. Scattered

Figure 1: Schematic diagram of experiment.

atoms are measured in the detector {x^z) plane

located a distance L from the interaction region

perpendicular to the atom beam (t/-axis).

Using a laser frequency tuned to excite only

F—2 ground state atoms to the 3^P state, the

photon-recoiled atoms are spatially separated

from the atoms that are in the off-resonant F=l

ground state. The displacement d of the photonrecoiled atom beam in the detector plane^'^ is re- lated to the excited-state fraction / of the atoms

in the interaction region by (two-level atom in

resonant laser field, rate-equation approach^'^)

where A is the wavelength of the resonant laser

radiation, r is the lifetime of the excited states,

M, V is the atom mass and speed respectively, h

is Planck constant and / is the length (along y- axis) of the interaction region. In order to obtain

absolute values of excited state cross sections it

is necessary to obtain reliable values of /.

Both circularly polarized and linearly polarized radiation were used. The linearly polarized

light had its electric vector parallel to the atom
84

UNPOLARIZED ELECTRON BEAM

UNPOLARIZED ELECTRON BEAM

Figure 2: Sodium atom 3P electron charge

clouds (from Ref. 11).

beam initial velocity direction (t/-axis). The

excited-state charge clouds prepared by the two

laser polarizations, shown in Fig. 2, correspond®

to different combinations of 3^P3/2,i^=3 state magnetic sublevels. No analysis is made on the

fined atomic state. Observations were mainly restricted to detector displacements parallel to the initial electron

momentum, mv. Referring to Figure 1, the atom

deflection in the detector plane due to superelastic and elastic or inelastic electron scattering

result in momentum transfer to the atom beam

counter to and along mv, respectively.

Kinematic analysis^ shows that under the condition mv -C MV, the x,z coordinates of the

scattered atom in the detector plane are related

to the polar and a^iimuthal angles of the scattered

electron as

X —- LPk sin 0 sin <f> (2) z = L{a-j3kCOse) (3)

where a ' mv/MV and fSk = mvk/MV. It is

Figure 3: Projection of momentum spheres in the

detector plane.

assumed that the atom is initially in the state £j

(we omit the subscript j from the initial atom

speed v). The subscript k refers to an electron

which after scattering leaves the atom in state

k, where £k is the final atomic state. This re- sults in elastic (sj = e^), inelastic [ej < £/.), or superelastic {sj > £k) scattering. Due to total

momentum conversation in the interaction, elas- tically scattered atoms will fall on a momentum

sphere with radius mv, while the radius mvk = mv corresponds to inelastically and

superelastically scattered atoms. Here E and

AE are initial electron energy and the energy

difference between initial and final atomic states,

respectively. These momenta spheres projected

onto the detector plane in configuration space,

shown in Figure 3 for 3 eV electron scattering,

correspond to circles with appropriate radii. The

elastic scattering radius is mvL/MV = aL in the

detector plane with the center at z = aL. When

the detector is placed inside a particular circle

it can detect atoms scattered into channels that

correspond to all momentum spheres of equal or greater radius.

The general case of the scattering process in electron-atom recoil experiments is described elsewhere^, where relations between two-
85

dimensional experimental parameters and the

differential cross sections cr{d) are quantitatively

discussed. Here we recall the final relations of

these analyses in order to proceed with a scatter- ing analysis for excited species. For that purpose

we define the position of the detector center to

be Z£, and consider oidy the case outside of the

fiill dc beam shadow when aU correction terms

to the scattering intensity can be neglected. The

scattering atom current /^(^x)) is then

(4)

where linizo) and louti^D) a-re scattering contributions into and out of the detector. Taking into

account all distribution functions of the real ex- periment, Iin{zD) is defined as

linizD)

X

^o/o(0)

2hAxAz

V{v)

/ E{E)dE

JEo-0.5

(5)

/ -^dv / Z{z)dz / X{x)dx

Jo V Jzi Jxi

j a{e)smede / d(i>

where Iq is the total electron number current,

/o(0) is the atom beam current at zjo — 0, h is the height of electron beam, Ax and Az are halfheight and half-width of the detector, "^{E) is the

energy distribution of the electron beam, V{v) is the velocity distribution of the atom beam, X{x)

and Z{z) are vertical and horizontal beam profiles normalized to unity at the beam center, Xi,

2:25 ^1, Z2 are limits of integration of atom beam

profiles, and

^1

4>2

a ZD — z —

COS (

a ZD — z -\- Az.

= — sm

_j Ax + X

PkL sin 9

'

. _i Ax — X

sm

(3kL sin 9

(6)

(7)

(8)

(9)

Outside of the dc beam scattering-out contributions are generally neghgible. However, this

process may contribute as a consequence of a

long "tail" of the dc beam in the region where

scattering-in is measured. This loutizo) contribution can be described with the simpUfied equation

ioIo{zD)(^T

loutizn) =

hV (10)

where Io{zd) is the atom beam current at Z£), F is the average atom speed and cry is total scattering

cross section.

The above equations describe the complete re- lationship between the scattering signal and the

scattering cross sections for a particular channel in an atom-recoil experiment. We will now

discuss under what conditions differential cross sections can be determined from such an experiment, first describing conditions of elastic scat- tering observation where small angle scattering

is not included. The second part is related to su- perelestic or inelastic scattering where small angle as weU as zero- angle collisions are included.

Assume that the influence of the electron en- ergy distribution on <7{9) through the limits of integrations of 0 and 0 is negligible^ in the present

electron energy range. Also, if the cross section is a slowly varying function of energy over the elec- tron energy distribution then the nominal peak of

that distribution is a good representation of elec- tron energy, and the integration over that distri- bution is unity. With regard to the atom velocity

distribution^'^ it is safe to take the average atom

speed, when medium range angle scattering is considered.

Due to finite widths of the detector and the

dc beam the detected signal is related to an

integrated differential cross section over certain

ranges of 9. Since 9 depends nordinearly on z (Eq. 3), when the detector is further to the right

of the dc beam the overall resolution is better.

The differential cross section can be taken out of

the integrations in the Eq. (5), assuming there is no rapid change of with respect to the 9 integration over the detector width and the changes

of (j) through the x and z integrations over the

beam widths. These conditions for elastic scat- tering are satisfied at the detector position cor- responding to 9 larger than, say, 20° so that Eq.
86

(4) can be simplified^ and the differential cross section in terms of the scattering signal is

1 hV

^(^) = r...r.r.A —U^D) + /o(^d)H (H)

7o(0)F(^)^ to

where

(zi3 + A2-z)

and

^e,x) = 2(sin

.1 Aa;

aL sin ^

a; . 1 Ax -\- X . . „, + sin-^ , . J.(13)

aiv sm 0

Eq. (11) is generally valid for both ground

state and excited state scattering processes but

only outside of the dc beam. In practice its apphcation is limited to the scattering angles cor- responding to where the detector would start seeing 0° angle scattering for inelastic processes,

in which case the scattering signal would corre- spond to the combination of both processes.

In the current experiment electrons are scat- tered by an incoherent mixture of ground- and

excited-state atoms, the latter constituting a

fraction / of the beam. Thus, the detected signal

is

I,{zn) = lf{zD)^in2D) 3Pl

where

io{l-f)

hV

(14)

(15)

x[Io{0)F{e)a''{6)-To{zD)4"]3Si

and

iof

hV (16)

x[Io{O)F{ey{0)-Io{zD)4'3Pl

The differential cross section for the excited-state

elastic scattering is

a'^{e):^j[cr%9)-{l~fy'{e)] (17)

where <r*{d) can be obtained from Eq. (11) with

(t{6) and ctt replaced by cr*{0) and and F{6)

is calculated using the laser-deflected atom beam

profiles. Here erf is defined as

(18)

where (jy^ and cr^^ are the total 3S- and 3P-state

scattering cross sections, respectively.

Superelastic scattering has an unique deflection signature, but the explicit form for the dif- ferential cross section cannot be derived since its scattering signal corresponds to small-angle scat- tering and the above approximations do not hold.

In this angular range (t{$) changes rapidly with

respect to 0 integration over the detector as well

as the beam width, and it cannot be removed

from the integrals in Eq. (5). In addition, ac- curate knowledge of the electron energy distri- bution and, even more important, atom beam

velocity distribution are required. Scattering signals (Eq. (4)) for superelastic colhsions contain an Iin{zD) contribution, described with the full Eq. (5) when the right hand side is multiphed

by / and an Iout{zD) contribution described by

Eq. (10) when ax is replaced by erf. To obtain

cr{B) for superelastic scattering we express-^" Eq.

(5) in matrix form with separated variables

h[zD) = A{zD,e)x{e) (19)

where h^zo) is the scattering signal, A{zd^6) in- corporates all functions included in Eq. (5) ex- cept cr(^), which is represented by the vector

x[6). Due to the fact that the scattering signal in

the range of measurements contain contributions

from collisions in the range 0° to about 20°, o-[6) can be analytically described by a function with

three independent parameters. The set of parameters that produces the vector h{zjj) which is in best agreement with the measured scattering

signal in the full range of observation defines su- perelastic a[9) in the appropriate angular range.

From higher level excitation (3^P -+ 4^5 and

3^D) measurements, only the tendency of

the cross sections can be extracted, due to resolution limitations even at low impact energies. For
87

2-

Na 3eV

3P- 3P

lineorly polonzed

loser light

3P-^3P o

circulorly polonzed

loser light

3S^3S

25 30 35 , ,40 B (degrees)

circularly polarized loser light

r.°0 o No

3S <—3P O lOev Q

O

•

• o o

-

•

•

•

•

Lj 1 1 1

1 o

1 1 1 1 .

-2 M

I (mm) -30.0 -22.5 -15.0 -7.5

Figure 4: Elastic scattering intensity, [zd con- verted into 9)

example, scattering signal at 3 eV contain contribution of 3^5 and 3^P elastic scattering from 35°

to 50° when small angle excitation peaks appear.

Those peaks are partially overlapped mainly due

to the atom beam velocity distribution.

Azimuthal angle dependency (Eq. 2, Fig. 3) can be studied by displacing the detector along

the x-ajcis (changing xb) at a fixed position of z (fixed 0). Prehminary data have been obtained

for elastic ground-state scattering and elastic

scattering by a superposition of ground-state and

3P-state sodium atoms. Observations reveal significant differences between the shape of the scat- tering signals for these two processes, but a quantitative analyzes will require additional effort. Examples of the measured scattering signals are given in Figs. 4 and 5 for elastic and su- perelastic scattering, respectively. All experimental functions included in Eq. (5) are carefully determined. Particularly important, the

atom beam velocity distribution is measured-^" by

laser-induced Doppler-shifted fluorescence with

Figure 5: Superelastic scattering intensity.

absolute frequency caUbration made by comparison with the zero-field hyperfine separation of

the sodium atom. Alternative measurements^ of

displacement of the forward scattering peak after

excitation by electron impact yield a consistent

distribution. Evaluated elastic and superelastic

cross sections yield information about partially state selective scattering ampHtudes, which complements measurements of charge cloud alignment and orientation by other experimental

groups

11

Acknowledgments

This work is performed in collaboration with

Prof. B. Bederson, M. Zuo, and T. Y. Jiang.

Research is supported by U. S. National Science

Foundation.

References

*Permanent Address: Institute of Physics,

P.O.Box 57, Belgrade, YU

1. B. Jaduszliwer, P. Weiss, A. Tino and B. Bederson, Phys. Rev. A 30, 1255 (1984).

2. K. Rubin, J. Perel, and B. Bederson, Phys.

Rev. 117, 151 (1960).
I

88

3. L. Vuskovic, M. Zuo, G. F. Shen, B. Stumpf

and B. Bederson, Phys. Rev.A 40, 133 (1989).

4. B. Jaduszliwer, G. F. Shen, and B. Bederson,

Phys. Rev.A 33, 3792 (1986).

5. R. Blatt, W. Ertmer, P. ZoUer, and J. H. HaU,

Phys. Rev.A 34, 3022 (1986).

6. I. V. Hertel and W. Stoll, in Advances in ( Atomic and Molecular Physics, edited by D.

R. Bates and B. Bederson (Academic, New York, \ 1977), Vol. 13, pp. 113-228. '

7. N. D. Bhaskar, B. Jaduszliwer and B. Bederson, Phys. Rev. Lett. 38, 14 (1977). I 8. M. G. Fickes and R. C. Stern, J. Chem. Phys. J 60, 4710 (1974).

9. M. Zuo, T. Y. Jiang, L. Vuskovic, and B. Bederson, Phys. Rev. A, submitted for pubUcation, I 1989.

10. T. Y. Jiang, M. Zuo, L. Vuskovic, and B.

Bederson, Phys. Rev. A, to be submitted, 1989.

11. N. Anderson, J. Gallagher, and I. Hertel,

Phys. Rep. 165, 1 (1988).
89

THE EFFECTS OF A FINITE SCATTERING VOLUME ON THE DETERMINATION OF

ELECTRON IMPACT COHERENCE PARAMETERS

P. W. Zetner*, S. Trajmar* and G. Csanakt

*Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA

t Los Alamos National Laboratory, Los Alamos, New Mexico, USA

1. Introduction

In electron-atom beam-beam scattering experiments, the interaction region defined by the intersecting beams and the viewcone of the detector should

be small in comparison with other characteristic di- mensions of the scattering geometry and the energy

and angular resolution of the apparatus should be nar- row with respect to the ranges over which quantities

of interest undergo significant changes. In this ideal

case, the scattering signal can be assumed to originate fi-om a point-like source and the data obtained

from the measurements can be assigned to well-defined

electron impact energy (E^,) and scattering angle (^e)- In practice, the dimensions of the scattering volume

and the energy and angular resolution of the apparatus are always finite and a rigorous treatment of the

scattering data should take this into account. In conventional electron scattering measurements of the dif- ferential scattering cross section (DCS) these considerations, in general, cause no serious problems. The

DCS can then be associated with nominal scattering

angles and impact energies and represents a sum over

final and average over initial experimentally indistinguishable processes. A discussion of these matters in electron scattering DCS measurements has been given,

for example, by Brinkmann and Trajmar^.

In recent years, a large body of data has accumulated in the field of electron impact induced orientation

and alignment of atomic valence shells^. Coherence

and correlation experiments used to examine these as- pects of the coUision process represent a considerable

refinement over conventional DCS measurements. In

principle, only the scattering signal originating from an ensemble of excited atoms prepared in a well-defined

quantum mechanical state is measured. The selection

of this ensemble is carried out either by coincidence

detection of the inelastically scattered electron and

emitted photon, or by laser preparation of the excited

state atomic target and detection of the superelastically scattered electron. In both cases, a photon inci- dence vector and polarization vector are defined v/hose

directions with respect to the scattering plane must be

well-specified in order that unambiguous conclusions can be drawn about collision induced alignment and

orientation (or, equivalently, about the electron impact coherence parameters, EICP). A rigorous evaluation of scattering data from coherence and correlation

experiments should, therefore, include the convolution

of the scattering angle as well as the photon direction

and polarization angles with the finite angular resolution of the apparatus. Generally, however, the picture

of an ideal point-like scattering has been applied to the

interpretation of coherence and correlation data.

A recent study by Martus et al.^ does take into

account the effect of averaging over the finite range of

unresolved scattering angles observed by the electron

detector in a measurement of the PI polarization cor- relation parameter. Here, we describe a more rigorous

approach to the problem of determining the influence

of a finite scattering volume in electron-photon coincidence experiments and in measurements of superelastic scattering from laser-excited atoms. We treat the interaction volume as an ensemble of individual

scattering points to each of which is attached a col- lision coordinate frame that may differ significantly

from a laboratory-fixed coordinate frame. A "true" scattering plane, defined by the incident and outgoing electron momentum vectors of a collision event oc- curring somewhere within the ensemble can be quite

different from the "nominal" scattering plane defined

in the laboratory frame. The transformation of coor- dinate frames has extremely important consequences

for the measurement of EICP and is unaccounted for by averaging over the scattering angles alone. Photon

incidence and polarization vectors which are specified

in the laboratory coordinate frame by spherical angles

6^ and (f>„ and by polarization angle are given in the collision coordinate frame by angles 9^, 4>n and tp
90

which can, depending on the nominal scattering angle,

vary radically for collision events occurring throughout

the extended scattering volume. This radical behavior is expected for nominal scattering angles close to zero since the rotation of a collision frame relative to the laboratory frame for a scattering event displaced

from the "nominal" scattering plane can be severe at small scattering angles. In this situation, the angular

resolution of the system is no longer narrow with re- spect to the range over which the photon polarization vector and the photon direction vector (in particular)

undergo significant variation. The extraction of EICP

at these scattering angles becomes meaningless in the

single-point scattering picture. Surprisingly, we have

also discovered that a scattering volume of finite extent can have a dramatic influence on the measurement

of EICP at scattering angles far from zero. In these

cases, the range over which the photon direction and

polarization vectors vary can be small compared to the angular resolution but the coincidence or superelastic scattering intensity becomes extremely sensitive to these small variations due to the behavior of the

EICP themselves.

In this paper, we specifically examine the conse- quences of a finite scattering geometry on "in-plane"

coherence experiments, i. e. polarization correlation

or superelastic scattering experiments in which the

incidence vector of the detected photon or the laser photon lies in the nominal scattering plane {(i>^ — 0).

The "in-plane" experimental configuration is adopted

in order to give direct information about spin-orbit

coupling effects. We show that a finite scattering geometry can give

rise to severe distortion in the dependence of the co- incidence or superelastic scattering signal intensity on

the photon polarization and that this distorted behavior can be misinterpreted as arising from spin-orbit

coupling effects.

2. Model Calculations

We have carried out modelling calculations in which the finite scattering volume is approximated by

a distribution of discrete points. At each point, a collision coordinate frame is defined in which the theoretical formalism of da Paixao et al.^ (also Ref. (5))

is applied. This formalism can be used to describe

electron-photon coincidence signal intensity for a J =

0 to J = 1 excitation or superelastic scattering signal

for a J = 1 to J = 0 deexcitation. For the case of

linearly polarized photons:

/ ~ A + B'cos2ip + B"sin2xp (1) where the coefficients A, B' and B" have been given**'^ as functions of the EICP and the photon incidence vector direction angles 0^, <f>n measured in the colli- sion coordinate frame. Implementation of the model

involves carryingout the coordinate frame transformation associated with a scattering center located at position vector fj from the laboratory frame origin. The

relation between the photon angles d^,, 0^,, tp^ specified in the laboratory frame and the angles 9^, (f>Tij ^

in the collision frame is thereby determined as is the

relation of the true scattering angle, Oe, to the nominal

scattering angle, 0°. Fig. 1 presents a comparison between the collision frame associated with a scattering

event located at position vector fj and the collision frame associated with an "ideal" scattering event taking place at the origin of the laboratory frame {fj =

0). This latter scattering event defines the nominal

scattering plane and the nominal scattering angle, 6°. The collision frame associated with the off"set scatterer

is clearly rotated with respect to the laboratory frame

as evidenced by the direction of the vector Y^^^ (normal to the true scattering plane) compared to 1^°^ =

Yiab (normal to the nominal scattering plane).

Fig. 1 Electrons emitted by the source (G) are scattered into detector (D). Collision coordinate

frames associated with two different scattering

events, taking place within the extended interaction volume, are shown (see text).

In our model, a weighted average of the scatter- ing signal intensity over the distribution of scattering
91

centers is taken according to:

where /s is the average superelastic or coincidence in- tensity and aij are weighting factors discussed below.

The signal intensity arising from a particular collision

frame is denoted Ij{ky) since (for an assumed parallel incident electron beam) this frame is completely

specified by the displacement vector fj, the nominal

scattering angle, &° and the momentum vector, k*^, of an electron scattered into the detector viewcone. The

summation over index i in Eq. 2 represents an approximation to the effect of finite angular resolution in the

electron detector. The contribution from all discrete

scattering points chosen to represent the extended vol- ume is calculated in the summation over index j. The

weighting factors, Cj-y, represent the effect of the spatial

intensity distribution of the incident electron beam,

the spatial response of the electron detector and the

behavior of the DCS over the range of scattering angles

defined by the extended scattering volume.

The sophistication of the model can be greatly in- creased to include, for example, the finite acceptance

angle of the photon detector (or the divergence of the

incident laser beam in the superelastic scattering case),

the density distribution in the target gas beam and so

on. We have found, however, that a relatively crude

description of the scattering geometry can provide the

essential aspects of the finite volume effect. In the re- sults presented here, an array of 5 points, equidistantly

spaced along a segment of the Yiab axis, was used to represent the extended scattering volume. The effect of the detector viewcone was simulated by associating

5 scattering directions with each of the 5 scattering

points in the linear array. The weighting factors, a,j-, reflect only the spatial intensity distribution (Gaussian) of the incident electron beam.

3. Results

3.1 Superelastic Scattering from Laser-Excited

"8Ba(-..6s6p ip)

Efforts to measure EICP for the ^^^Ba ^So to ^Pi

transition in a superelastic scattering experiment have

been reported by Register et al.^. An experimental

configuration was adopted in which the laser beam lay

within the (nominal) scattering plane and the degree of

modulation of the superelastic intensity, !« , as a function of laser beam polarization angle, rp^,, was measured. During the course of their investigation, Register et al. discovered that this modulation, Ia(V'i/), was, unexpectedly, asymmetric with respect to xjji, 0 where V*!/ = 0 is defined to be the angle at which the

laser beam polarization vector lies within the scatter- ing plane. Furthermore, this asymmetry was scatter- ing angle dependent in a regular way and was found to be reproducible despite major variations in experimental conditions. The origin of this asymmetry remained

unexplained and EICP could not be extracted, with

confidence, from these measurements. This led us to reinvestigate the cause of the asymmetry in more detail, both experimentally and theoretically, with special emphasis on the effects of a scattering volume of

finite extent.

Superelastic scattering data can be straightforwardly analyzed by fitting Eq. 1 to the observed Is'y^pv) modulation curves. For this purpose, Eq. 1 is written

in the equivalent form:

Isi'^i.) ^l + r) cos[2^j}^^-2a) (3)

where r; is the modulation depth and a is the modulation phase shift.

For a laser beam lying in the scattering plane,

theory^'^ predicts that a = 0. However, the asymmetry observed by Register et al. and by us, in a more recent series of experiments, can be represented

by a modulation phase shift which is non-zero. Fig. 2 shows the values of "q and a extracted from our recent superelastic scattering measurements and the results of the corresponding model calculations for incident

electrons of SOeV impact energy. These model calculations employed the theoretical EICP of Clark et al7

which indicate that the ^^^Ba(- • -GsGp ^P) state is al- most purely LS coupled. It is important to note that,

for a purely LS coupled state, an ideal single atom col- lision occurring at the origin of the laboratory frame

(for laser incidence angle <f>i, -— 0) would give r] = 1 and

a = 9 at all scattering angles. Fig. 2 illustrates that

serious deviations from these predicted values occur

in an actual experiment and that these deviations are explained by the extended scattering volume model.

The model calculations indicate that, for a laser beam incidence vector lying in the nominal scatter- ing plane, the phase shift differs from zero when

the extended scattering volume is asymmetrically dis- tributed in the normal direction to the nominal scat- tering plane. We have also verified experimentally that

the deviation of a from zero is tied directly to asymmetry in the distribution of excited-state scatterers^'^.
92

The distortion in the modulation depth, however, persists regardless of how the scattering volume is dis- tributed. These results imply that the phase shift problem could conceivably be eliminated by ensuring a

synunetrically distributed scattering source. However,

EICP are extracted from the modulation depth, t], which suffers significant distortion whenever the scat- tering volume is finite in extent.

1

Q.

lii Q

Z

O

3

QO

-40 -20 0 20 SCATTERING ANGLE (deg)

100

Xw

UJ£0<

X

Q.

-40 -20 0 20 SCATTERING ANGLE (deg)

Fig. 2 Model calculations and experimental data** for superelastic scattering from ^^^Ba(' • -GsGp ^P) at 30 eV. Laser beam incidence angles are $i, — 90°; = 0°. (a) Modulation depth vs. nominal scat- tering angle; (b) Pheise shift vs. nominal scatter- ing angle.

Fig. 2 also reveals that the distortion in modulation depth produces relatively well-defined features

which appear at specific scattering angles. For the

superelastic scattering process under discussion, the

scattering angle location of these features is closely tied to the behavior of the alignment angle, 7, of the excited

state charge cloud produced by the time-inverse inelastic scattering process. When this alignment angle is such that the laser photon incidence vector lies along

the major axis of the charge cloud (assuming a colli- sionally excited "p" type orbital) then the measure- ment becomes extremely sensitive to extended volume

effects^. Although it is not apparent from the figure,

we have also found that the severity of the geometryinduced distortion is related directly to the shape of

the charge cloud produced by the time-inverse inelastic process. In particular, when the ratio of length

to width of this charge cloud is large at the "critical" alignment angle described above, then the distortion

in r} becomes severe. The smaller this ratio, the less sensitive the measurement becomes to extended vol- ume effects. Of course, this ratio is given by the P^

coherence parameter and we claim that, when P^ is close to unity at the critical aUgnment angle, one can

expect a strong influence of geometry on the measure- ment of T/.

3.2 Measurements of the P4 Coherence Parameter in the Rare Gases

Conclusions about the influence of an extended

scattering geometry on "laser-in-plane" superelastic

scattering experiments are directly transferrable to studies concerning the P4 Stokes parameter in rare %z&

J =: 0 to J = 1 excitations. We, therefore, extended

our modelling effort to the rare gases and, in Figs.

3 through 5, the results are compared with the only

P4 measurements so far available in the literature®.

The same crude model was employed as in the case of superelastic scattering from ^^®Ba(- • -GsBp ^P). The

photon detector was placed in the nominal scattering

plane at 90° relative to the incident electron beam and was assumed to have infinite angular resolution. The

EICP of Bartschat and Madison^° were used.

Fig. 3 shows the P4 vs. scattering angle behavior

for the ^P state excitation in Ne. Dots (crosses) rep- resent modelling calculations carried out with a finite scattering volume of 1mm (2mm) extent. The ^P state

is described by pure LS coupling so that P4 is expected

^ to be unity over all scattering angles. Near zero de- 5 grees scattering angle, the influence of a finite scatter- ing volume reduces the P4 polarization to a minimum

of about 0.8.

For the heavier rare gases, Kr and Xe, we found

that, as discussed above, the shape of the modelled P4
93

vs. scattering angle curve was quite sensitive to the

behavior of the theoretical near the "critical" alignment angle. The model calculations plotted in Figs. 4

and 5 result from slightly modifying the behavior

predicted by Bartschat and Madison. The calculated

behavior of the aUgnment angle was left unmodified.

5 10 15 20 25 30 35 SCATTERING ANGLE (deg)

Fig. 3 Model calculations and experimental data® for

excitation of Ne 3s'[l/2]5 ^Pi at 80 eV impact

energy.

We performed this exercise in order to check

whether the experimental P4 data, whose deviation

from unity is ascribed to spin-orbit coupling eflPects®, could be fit by adjusting only the theoretically pre- dicted behavior of P^ which is not associated with

spin-orbit couphng. The fit of the model results to the

experimental data in Figs. 4 and 5 is reasonable, suggesting that, under conditions where spin-orbit coupling efiects are negligible (i.e. cos e = 1 or poo = 0),

extreme distortion in the P4 measurement can occur

for a particular behavior of the coherence parameters

P^ and 7 (or, equivalently, A ajid x) ^ scattering

volume of finite extent. The existence of an extended

scattering geometry is a necessary condition for this

extreme behavior in P4 to manifest itself. This is il- lustrated by the fact that, for an ideal, single-point

Fig

0, 5 10 15 20 25 30 35 SCATTERING ANGLE (deg)

, 4 Model calculations and experimental data® for excitation of Kr 5s' [1/2]? ^Pi at 60 eV.

11

DC

LU

HLU<

<

Q.CO

111 O

Fig

5 10 15 20 25 30 35 SCATTERING ANGLE (deg)

, 5 Model calculations and experimental data® for excitation of Xe 6s[3/2]$ *Pi at 30 eV.
94

scattering at the laboratory frame origin, P4, in the

case of Kr, is expected to be unity over the range of

scattering angles studied, while, in the case of Xe, the

dotted line in Fig. 5 shows the predicted behavior of

P4.

These results will be discussed in more detail in a forthcoming publication^^.

Conclusions

Results of our modelling effort imply that measurements of EICP using experimental geometries in which the photon incidence vector (either laser photon

in the superelastic scattering experiment or detected

photon in the polarization correlation experiment) lies in the nominal scattering plane can be subject to large

uncertainties. The influence of an extended scattering

volume can produce severe distortion in the measured

modulation depth, rj, or Stokes parameter, P4, which

can be misinterpreted as arising from spin- orbit coupling effects. The severity and location (in scattering

angle) of this distortion is intimately tied to the behavior of the P£" and 7 parameters (or, equivalently,

the A and x parameters).

Acknowledgements

This work was supported by the National Science

Foundation, the National Aeronautics and Space Administration and the U.S. Department of Energy.

References

8. P. W. Zetner, S. Trajmar, G. Csanak and R. E. H. Clark, Phys. Rev. A 39, 6022 (1989).

9. P. Plessis, M. A. Khakoo, P. Hammond, J.J. Corr

and J. W. McConkey, J. Phys. B: At. Mol. Opt.

Phys. (in press, Oct. 1988).

10. K. Bartschat and D. Madison, J. Phys. B: At. ^ Mol. Phys. 20, 5839 (1987).

11. P. W. Zetner and S. Trajmar (to be published).

1. R. T. Brinkman and S. Trajmar, J. Phys. E. 14,

245 (1981).

2. N. Andersen, J. W. Gallagher and I. V. Hertel,

Phys. Rep. 165, 1 (1988).

3. K. E. Martus, K. Becker and D. H. Madison, Phys.

Rev. A 38, 4876 (1988).

4. F. J. da Paixao, N. T. Padial, G. Csanak and K.

Blum, Phys. Rev. Lett. 45, 1164 (1980).

5. P. W. Zetner, S. Trajmar and G. Csanak (to be

published).

6. D. F. Register, S. Trajmar, G. Csanak, S. W.

Jensen, M. A. Fineman and R. T. Poe, Phys. Rev. A 28, 151 (1983).

7. R. E. H. Clark, J. Abdallah, Jr., S. P. Kramer and

G. Csanak, Phys. Rev. A. (to appear).
95

SPIN DEPENDENCE IN ELECTRON IMPACT IONIZATION OF ATOMIC HYDROGEN

M.S. LUBELL, D.M. CROWE, X.Q. GUO, F.C. TANG, and A. VASILAKIS

Department of Physics, City College of CUNY, New York, NY 10031, USA

J. SLEVIN

Department of Experimental Physics, Sl Patrick's College, Maynooth, Co. Kildare, Ireland M.EMINYAN

Laboratoire de Physique Atomique, Tour 24, Universite Paris Vn, F-75251, Paris, France

Using beams of polarized electrons and polarized hydrogen atoms, we investigated the spin dependence

in impact ionization from threshold to 500 eV. In order to provide greater insight into threshold laws

for double escape, we devoted considerable attention to the energy region from 13.6 to 15.3 eV. Our

measurements of the cross-section asymmetry, Ai, for electron spins antiparallel and parallel display

inconsistencies with a naive application of the Wannier model. Moreover, they suggest the presence

of structure, the general features of which can be reproduced by a fit of the Coulomb-dipole model. At

higher energies, our asymmetry measurements appear to scale with the incident electron energy, E, according to the relation Ai a E"^-^. .

1. INTRODUCTION

In the context of electron-atom scattering, the electron-hydrogen collision has served for many years as one of the principal testing grounds for the development of theoretical approximation techniques. To great accuracy, the low-energy electron-hydrogen problem can be treated in a non-relativistic framework, as a consequence of which the spin-averaged scattering cross section a can be expressed as^

and exchange amplitudes respectively. As can be seen from Eq. (1), the experimental determination of the spin-averaged cross section falls short of providing detailed knowledgeof the individual amplitudes which are caculated theoretically. As a first step toward achieving the finer detail required, we have employed polarized

beams to measure the spin-dependent ionization cross-section asymmetry Aj defined by

where a(Ti) and a(TT) are respectively the angle-integrated ionization cross-secuons for incident and

atomic electron spins antiparallel and parallel. From an analysis of spin-tagged scattering, it can be shown

that Ai is given by^

A = o(Ti) -a(TT)

c(T>l) +a(TT) (2)

A |f||g|cose/5

(3)

where 9 is the relative phase between f and g. Alternatively Aj can be expressed as

A

1 - r

1 + 3r (4)

where r is the ratio of the triplet to singlet cross sections.
96

The improved insight into the process of impact ionization provided by measurements of Aj can

already be seen from exploratory studies of electron-hydrogen scattering carried out several years ago^ with

relatively primative sources of polarized electrons and polarized hydrogen atoms. Those studies, which

utilized a Fano-effect electron source and a thermal-dissociation hydrogen source, revealed previously

undocumented deficiencies in virtually all theoretical models at incident electron energies below -60 eV. Moreover, the apparent non-unity value of Aj at the ionization threshold called attention to the theoretical misconception^ that all triplet states should be supressed at threshold.^ Similar studies carried out for 90°

elastic scattering and for a variety of scattering channels for the alkali atoms'^ have proven to be equally

valuable.

2. EXPERIMENTAL METHOD

The general methodology of our present experiment follows that of the earlier work^ in that the quantity we measured was the ionization counting-rate asymmetry, Ai, defined by

R(ti) -R(TT)

R(Ti) +R(TT)

(5)

where R(Ti) and R(TT) are the ionization counting rates, normalized to incident beam intensities and

corrected for residual gas effects, for the respective antiparallel and parallel electron spin configurations. The

quantities Ai and Aj are related by the expression^

(6)

where Pg and Ph are respectively the electron and hydrogen polarization vectors, a is the angle between

them, and F2 is the fraction of events attributable to hydrogen molecules. In a separate study^ we

determined that Ph had the value 0.515(5), and from a normalization of our new ionization asymmetry

results to our older ones we determined that Pg had the value 0.27(2). On the basis of magnetic field studies

in the interaction region, we found that we could take Icosal to be 0.995(5). The fraction of events

attributable to hydrogen molecules, of course, depended upon the electron beam energy, but with a

dissociation fraction of -0.95 characterizing the hydrogen beam in the interaction region, F2 never exceeded

0.3 over the entire energy range investigated. The residual gas pressure in the interaction was

approximately 5x10"^ Torr. Our experiment employed a crossed-beams geometry, as illustrated in Fig. 1. Polarized electrons were

produced by photoemission from a <100> GaAs crystal^'^ with the use of 780-nm circularly polarized light

generated from a GaAlAs diode laser and prepared by a combination of a half-wave (1/2) and a quarter-wave

(1/4) retardation plate, as shown in the figure. Before reaching the interaction region, the extracted electron beam passed through a set of electrostatic electron-optical elements comprising a 90° spherical bender, a

series of lenses and steerers, and a 180° hemispherical monochromator, the overall geometric orientation of

which ensured that the electron polarization vector, Pg, was pointing either parallel or antiparallel to the direction of the hydrogen beam. The usable electron current, which varied between 30 and 150 nA, was

monitored for normalization purposes by a Faraday cup located just beyond the interaction region.

The polarized atomic hydrogen beam^ originated in an rf dissociation source and underwent high-field

state selection in a pair of permanent hexapole magnets which also served to focus the ms=-i-l/2 atoms at the interaction region. Upon emerging from the second magnet, the atoms entered a small solenoidal field which adiabatically rotated their spins into a longitudinal direction either parallel or antiparallel to the direction of the hydrogewn beam. Additional coils maintained the magnetic field in the longitudinal

direction and trimmed it to -100 mG at the crossed-beams interaction site. Prior to entering the interaction region, the hydrogen beam was collimated to a radial dimension of 3 mm, resulting in a usable average

target density of -10^^ atoms/cm^- For normalization purposes, the hydrogen beam was sampled at the end of its line by a quadrupole mass analyzer (QMA). For purposes of "real-time" background subtraction,

the hydrogen beam was equipped with a mechanical chopper located between the two hexapole magnets, as shown in Fig. 1.
97

26

(^^ (21)

"*—

^—|~"fl^''"B~0^''H]

1^ ®

Fig. 1. Schematic layout of the experiment showing (1) hydrogen source, (2) hexapole magnets, (3) hydrogen beam chopper, (4) solenoid, (5) hydrogen polarization vector, (6) electron monochromator, (7) ion

detector, (8) electron spectrometer, (9) solenoid, (10) adiabatic spin rotator, (ll)Stem-Gerlach polarimeter,

(12) QMA, (13) 780-nm laser light, (14) GaAs crystal, (15) 90° spherical bender. (16) solenoid, (17) electron

polarization vector, (18) Lyman-alpha detector, (19) Faraday cup, (20) exit to Mott polarimeter, (21) GaAlAs

diode laser, (22) lenses (23) 1/4-plate, (24) 1/2-plate, (25) laser beam flag, and (26) vacuum window. Item

(11), used for the measurement of Ph, was removed for the present studies as were items (8), (18), and (20).

For ion detection, a channel electron multiplier (CEM) was used, preceded by a grid which prevented

electrons or negative ions from reaching the CEM entrance cone. The grid and cone voltages were varied to ensure that the signal was free from photon contamination under the final operating conditions. In order to eliminate the influence of stray electric fields on the ions drifting downstream from the crossed beams site, the interaction region and the CEM were enclosed in a stainless steel box, the dimensions of which were

chosen such that a 4k acceptance was still presented to ions having maximum recoil energy.

In carrying out the asymmetry measurements, we sought to reduce systematic effects by reversing the

direction of the electron polarization at frequent intervals. To this end, under computer control, we rotated

the 1/2 and 1/4 plates through 360°, the former in 8 steps of 45° and the latter in 4 steps of 90°. During

data acquisition, we time ordered the retardation plate orientations to produce 8 helicity sets, each of which

contained 4 circular polarizations alternating between left and right. Thus we generated 8 sets of data in each of which Pg and Ph alternated between antiparallel and parallel configurations with each configuration

appearing twice. Several times we also checked for systematic effects by reversing the magnetic field in the

interaction region, thereby reversing the direction of PhFor each of the 8 data sets, (i), within a given run, we calculated a single "real" asymmetry, Ar(^)

from the average of the rates R(Ti)i(') and R(T>L)2(') for the two antiparallel configurations and the average

of the rates R(TT)i(0 and R(TT)2(') for the two parallel configurations. In order to remove systematic

effects associated with hydrogen beam instabilities and drifts in the QMA tuning, as well as spurious

helicity-correlaied behavior, we followed procedures similar to those used in the earlier work,^ and for each

of the data sets we generated two "false" asymmetries, Af±^'\ in accordance with the prescription

^1F±

(0 (i)" R(Ti) -R(n)

1 2 - ±

(i) R(Tt)

1

(i)" -R(TT)

2 -

(l) (i) 'R(Ti) +R(Ti)

I 2 - +

(i) R(TT)

1

(i) + R(TT)

2 - (7)

If any false asymmetry deviated from zero by more than 2.5 standard deviations (a 1% occurrence

statistically), we deleted the entire data set to which it corresponded, an approach that is somewhat

conservative but justifiable, given our goal of a ±2.5% relative uncertainty in the determination of Ai. Using this criterion, we deleted approximately 25% of our events, the vast majority of which failed the cut because of QMA tuning drifts during several isolated periods when the laboratory temperature rose to unacceptably high levels. Having made this cut, we found that each of the sixteen false asymmetries,

generated in accordance with Eq. (7), was consistent with zero at the statistical level of 7x10"'^ when
98

averaged over all energies examined. A slightly large value of 1.2 for a typical reduced chi-square for approximately 150 degrees of freedom, however, suggests the presence of some residual noise at a very low

level. In the course of our work, we found that the energy of the electron beam drifted slowly upward by as much as 50-75 meV during one e-folding lifetime of the GaAs crystal.^ We attribute this drift to the continuing change in the negative electron affinity crystal as the surface slowly degraded following each

activation. Since our electron source was typified by a lifetime in excess of 400 h, the energy drift was a

relatively mior problem. Nonetheless, in order to achieve a high degree of confidence, we found it useful

to calibrate the beam energy every 24 to 48 h by turning off the rf power in the hydrogen source and

measuring the molecular ionization rate at energies between 0.3 and 1.0 eV above threshold. For our

electron energy spread, characterized by a Gaussian halfwidth, a, of 75 meV, the convoluted ionzation cross section^ over this energy range is nearly linear, all the vibrational and autoionizing structure remaining

unresolved. We therefore were able to use a linear extrapolation of our measurements to an apparent

threshold value of 15.50 eV as a means of calibrating the energy of the beam to a typical accuracy of

±30 meV. We note that although the energy drift was substantial enough to require periodic calibration, it was sufficiently small that the polarization of the beam remained effectively unchanged. Length restrictions preclude a complete description of the method we used to determine the energy spread of the electron beam.

Suffice to say that we carried out a series of ionzation rate measurements of atomic hydrogen (source rf turned on) between 13.67 and 13.87 eV. From a study of the Wannier power-law^-^^ convoluted with

various assumed Gaussian energy distributions, we concluded that our beam was characterized by a Gaussian

halfwidth, a, of 75 meV, as indicated above.

In carrying out our asymmetry measurements, we accumulated data at different energies in a quasirandom sequence. In order to check for stabiltiy of operation, however, we frequently returned to the measurementof Aj at an energy of 15.07 eV. The results, shown in Fig. 2, indicate that our apparatus was

indeed stable at a level appropriate to the accuracy of the measurements. At incident energies above the 15.43 eV molecular ionization threshold, we had to correct our asymmetry measurements for events related

to molecules. We used a procedure similar to that employed in the earlier work^ and determined the fraction of events, F2, due to molecules with the aid of the QMA mass-2 signals when the hydrogen source rf was

on and when it was off. In the present work, we also had to make corrections for dynamic beam loading^ ^ of the interaction region because the hydrogen beam chopping frequency was relatively low

(-0.5 Hz). With these corrections made, we calculated the atomic ionization asymmetry Ai^^ according to

the general prescription Ai^^ = Aj / (I-F2), suggested by Eq. (6).

Fig.2. Stability check showing 24 separate measurements of Ajat 15.07 eV. The average value is 0.0682(5) with a reduced chi-square of 1.17 for 23 degrees of freedom corresponding to a confidence level of 27%.

3. RESULTS AND DISCUSSION <

Hi

In Fig. 3 we present our measurements of Aj ^ for incident energies, E, between threshold and 500

eV, only two points being shown below 15 eV for purposes of clarity. We note that for incident energies

above 15.1 eV, the asymmetry is represented quite well by the functional form Ai^^ = 0.47E'^-^, where E

is expressed in eV. In order to place our asymmetry measurements on an absolute scale, we used this functional form, convoluted appropriately, to normalize our results to the data of the earlier work.
99

Although any individual data point in the earlier work suffers in accuracy and energy resolution by contrast with its corresponding data point in the present work, the normalization procedure involves a sufficient number of measurements that a scaling accuracy of ±7% in Ai is achieved. The normalization of our asymmetry data is consistent with a value of 0.27(2) for the electron polarization, Pg, which although

somewhat lower than originally anticipated, is consistent with observations made at a number of other

laboratories for GaAs crystals operating at room temperature.

\

1 \ 1 1 1—1—r—I—] 1 1 1

1

1—1—1—ry

/

1 1 1 1 1 1 1 1

)V. I I 1 < I , , , , I i I I I I I I ; _J i » I I I I I 10 15 20 50 100 150 200 500 1000 ENERGY, E (eV)

Hi

FIgJ. Measured values of Ai ^ with enor bars representing one standard deviation uncertainties. The solid

line represents a constant value of 0.07 for the electron energy E below 15.1 eV and a power law given by

0.47E'°-^ above 15.1 eV with E expressed in eV. The hatched region at the left lies below the 13.602 eV

ionization threshold.

1.0

0.8

0.6 -

0,4

0.2

0.0

1 1

11 1

I

1 1 1 1

I

1 1 1 1 1 1

I

1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 M n 1

1 1 1 1 inn 1

1

1

\ - \

1 1

/

1 . , 1 , ,

^^^^

1 1 Ml, 1 1 1 1 1 1 1 1 1 ;i,Li ^~a"f^ ^

iMUl 1 i I. 15 20 30 50 100 150 200 ENERGY (eV) 300 500

Flg.4. Summary of the asymmetry measurements Aj from the present experiment, shown as closed circles, and those of the earlier work (Ref. 1), shown as open rectangles. In the latter case, the height represents a one standard deviation uncertainty (±a) while the width reflects the full-width-at-half-maximum energy spread. In the former case, the vertical error bar again represents a one standard deviation uncertaainty, but the energy width of ±75 meV is too small to be seen. The theoretical curves presented represent a sample from the

literature: (a) coupled-channel distorted-wave optical potential (Ref. 13), (bj) Bom-exchange (Ref. 14), Born- exchange with exchange maximized (Ref. 14), (c) Bom-exchange (Ref. 15), (d) Glauber-exchange (Ref. 16),

(e) pseudostate close-coupling (Ref. 17), and (f) modified Bom-Oppenheimer (Ref. 18).

Having placed our asymmetry results on an absolute scale, we can compare them with theoretical predictions. Fig. 4. provides such a comparison for a representative number of calculations, along with a

display of the data from the earlier work.^ We observe that one recent calculation by Bray, McCarthy, and

Madison^^ provides a rather good fit to the data over the full energy range down to about 20 eV. Only at
100

an energy of 40 eV, is there a significant discrepancy, and at this energy there is also a discrepancy between

the new and old data. We have no explanation for the disagreement between the new and old data, although we note that the "knee" suggested by the new data is reminiscent of a similar structure observed for the ionzation asymmetry in electron-potassium scattering.

We now turn our attention to the threshold region. In Fig.5 we present our measurements of Aj for incident energies, E, between 13.67 and 15.27 eV, as well as one measurement obtained with the centroid of

the beam energy at 13.57 eV, 32 meV below the laboratory threshold Eth= 13.602 eV. Within the/u//

region (Eth<E< 15.27 eV), our measurements of Aj are clearly inconsistent with a linear dependence on

energy. A linear fit in fact yields a reduced chi-square of 5.5 for 16 degrees of freedom corresponding to a confidence level of <10"^^. It therefore appears that some structure is present, with any presumed sharp

features masked by the ±75 meV spread in the energy of the electron beam. We note that if the false asymmetries are plotted as a function of energy, no similar structure is evident.

0.05

1 -T - I I 1

1 1 1 1 11 i

1

(b)

1

1 1 1 1 1 1 1 1 1 1 1 1 13.5 14.0 14.5 15.0 E (eV)

Fig. 5. Measured asymmetries Aj for incident electron energies between 13.57 and 15.27 eV. The vertical and horizontal error bars represent one standard deviation uncertainties. The heavy outer bar on the 13.57 eV

point illustrates the characteristic energy spread of the electron beam. The curve (a) is a constant function fit in the restricted (hatched) region from threshold (13.6 eV) to ~14.1 eV. The curve (b) is taken from the Coulomb-dipole theory (Ref. 19) with fitted model parameters as given in the text.

It is reasonable at this point to compare our results with the predictions of ionization threshold laws. A naive approach to the Wannier law predicts that if the ^S^ and ^P® final states are ignored^—the latter is inaccessible in our experiment in any case—Ai should be constant throughout a restricted region, which for hydrogen has been found to extend for about 500 meV above threshold. Although our ±75 meV energy

spread makes precise comparisons somewhat problematic, we note that a fit of a constant function to our

data within this region results in a reduced chi-square of 2.75 for 5 degrees of freedom, corresponding to a confidence level of 0.02. As a crude approximation, we can employ a linear function to test for the trend of

Ai in the restricted region. The resulting fitted function, having a slope of +0.019(7) eV"^, is characterized by a reduced chi-square of 1.77 for 4 degrees of freedom, corresponding to a confidence level of 0.17. The

notion that Aj falls with decreasing energy upon approach to threshold is reinforced by the presence of the 13.57 eV data point, which has an anomalously low value, albeit with a large uncertainty. Although this point lies slightly below threshold, it is almost unquestionably dominated by impact ionzation events produced by the high-energy tail of the electron distribution and lying within -50 meV of threshold. With

photon contamination of the signal ruled out experimentally, the only other contribution to the 13.57 eV

signal can arise from Rydberg-excited atoms with principal quantum number n>41 which can be field- ionized^^ in the vicinity of the ion detector. An analysis of ionization rates slightly above threshold places a strict upper limit of 50% on such a contribution to the 13.57 eV data point. Consequently, even if all of

the Rydberg contributions to the asymmetry improbably saturated at the lower bound of -i/3 for pure triplet excitation, 1 the value of Aj at 13.57 eV could still rise to only 0.065, thus confirming the positive slope

suggested by the data points above threshold.
101

We conclude with a brief comparison of our data to the predictions of the Coulomb-dipole model as developed by Srivastava and Temkin,^^ in which the partial cross sections near threshold obey the relation

a = C [ 1 + (D/a) sin (a In E + <[))]/[ In E' - X (9)

where E'=E-Eth and where the model parameters can take on different values for singlet and triplet

scattering. The Coulomb-dipole prediction for Aj follows from Eq. (3). From curve (b) in Fig.5, which

represents a fit of the prediction convoluted with our energy spread, it is evident that the Coulomb-dipole

model reproduces the general features of the data over the full energy range shown. However with Aj/Ai

taken as 0.124, the fitted set of model parameters, {Ct/Cs=0.321, as=27.8, at=12.1, Xs=4.58, Xt=4.92,

Ds=-2.22, Dt=1.06, <I)s=3.71, (|)t=1.80], results in a rather poor reduced chi-square of 2.68 for 8 degrees of

freedom, coresponding to a confidence level of -0.01. We hasten to point out, however that the Coulombdipole model may not he fully valid over the 1.7 eV range to which we applied it. Moreover, it is intriguing that the confidence level of the fit worsens by several orders of magnitude if the data points are

shifted in energy by 70 meV and the energy spread used in the convolution is increased by a factor of two. Nonetheless, it appears that new measurements with improved energy resolution are required if a definitive statement is to be made.

We gratefully acknowledge the financial support of the U.S. NSF, the Research Foundation of

CUNY, the U.K. SERC, St Patrick's College, NATO, and the French CNRS. We also wish to thank

Prof. A.R.P. Rau, Dr. A. Temkin, Prof. P.A. Souder, and Prof. K. MacAdam for a number of useful

discussions.

REFERENCES

1. G.D. Fletcher et al., Phys. Rev. A 31. 2854 (1985); T.J. Gay et al., Phys. Rev. A 26, 3664 (1982);

M.J.Alguard, et al., Phys. Rev. Utt. 39, 334 (1977). 2. H. Klar and W. Schlecht, J. Phys. B 9, 1699 (1976).

3 . C.H. Greene and A.R.P. Rau, Phys. Rev. Lett. 48, 533 (1983); J. Phys. B 16, 90 (1983). 4. See for example M.H. Kelley et al., Phys. Rev. Lett. 51, 2191 (1983); J.J. McClelland et al., Phys. Rev. Utt. 55. 688 (1985); 56, 1362 (1986); 58, 2198 (1987); J. Phys B 20, L385 (1987); Phys. Rev. A 40. 2321 (1989); G. Baum et al., L Phys. B 14. 4377 (1981); 18, 531(1985): Phys. Rev.

Lett 57. 1855 (1986); D. Hils et al., J. Phys. B 15. 3347 (1982); P. Nass et al., Z. Physik D 11. 71 (1989).

5. N. Chan a/., Z. Physik D 10. 393 (1988).

6. F.C. Tang et al.. Rev. Sci. Instrum. 57. 3004 (1986). 7. X.Q.Guo et al.. Rev. Sci. Instrum.. in press. 8. J.W. McGowan a/., Phys. Rev. 167. 52 (1969). 9. G. Wannier. Phys Rev. 90. 817 (1953); A.R.P. Rau, Phys. Rev. A 4. 207 (1971); R. Peterkop. L

Phys. B 4, 513 (1971); T.A. Roth. Phys. Rev. A 5. 476 (1971); H. Klar, J. Phys. B 16, 99 (1983). 10. J.W. McGov/an and E.M. Clarke, Phys. Rev. 167. 43 (1969). 11. M.S. Lubell a/.. Phys. Rev. A. submitted. 12. R.J. Celotta and G. Baum. private communications.

13. I. Bray et al., Phys. Rev. A. submitted. 14. R. Peterkop. Zh. Eksp. Teor. Fiz. 41. 1938 (1961) [Sov. Phys.-JETP 14. 1377 (1962)]. 15. S. Geltman a/., Proc. Phys. Soc. London 81. 375 (1963). 16. J.E. Golden and LH. McGuire. Phys. Rev. Lett. 32. 1218 (1974). 17. D.F. Gallaher. L Phys. B 7. 362 (1974). 18. V.L Ochkur. Zh. Eksp. Teor. Fiz. 47. 1746 (1965) [Sov. Phys.-JETP 20. 1175 (1965)]. 19. A. Temkin. Phys. Rev. Lett. 16, 835 (1966); 49. 365 (1982); J. Phys. B 7. L450 (1974); IEEE

Trans. Nucl. Sci. NS.30.1106 (1983); Phys. Rev. A 30, 2737 (1984); A. Temkin and Y. Hahn,

Phys. Rev. A 10. 708(1974); M.K. Srivastava andA. Temkin, Phys. Rev. A, submitted. 20. T.F. Gallagher, Rep. Prog. Phys. 51, 143 (1988).
102

PROGRESS ON SPIN DETECTORS AND SPIN-POLARIZED ELECTRON SCATTERING

FROM Na AT NIST

Jabez J. McClelland

Center for Atomic, Molecular, and Optical Physics, NIST, Gaithersburg, MB

Recent progress in the Electron Physics Group at NIST is discussed. Improvements

have been made on the low-energy diffuse-scattering spin analyzer, reducing instru- mental asymmetries and boosting the effective Sherman function. A figure of merit

of 2.3 X 10"'* has been achieved. Thorium has been used as a target in a 100 keV

retarding Mott spin analyzer, resulting in an effective Sherman function as high

as 0.49. This increased Sherman function, together with an increased scattering

intensity, results in a factor of 2 increase in the figure of merit. Good agreement

is seen between experiment and theoretical predictions of the Sherman function

for thorium. A hierarchical description of the T-matrix is discussed as a context

for interpreting recent results on spin-polarized electron scattering from optically

pumped sodium. Results are presented for elastic and superelastic scattering at 20 eV incident energy.

1 Introduction

In this paper, I would like to discuss three separate

subjects. Though they may seem somewhat unrelated,

they all pertain to the general theme of our work on

state-selected electron-atom scattering in the Electron

Physics Group at NIST. In the first two sections, I will present some recent work on the improvement of exper- imental techniques associated with the measurement of

electron spin. Innovations in the design of a compact

low-energy spin analyzer will be discussed, followed by a

description of some recent work on improving the Sher- man function of a Mott analyzer by using thorium as a

target. In the last section, I will consider some of our

latest state-selected results for 20 eV electron scatter- ing from sodium, with particular emphasis on how they

contribute to a complete picture of electron-sodium scattering.

2 Improved low energy electron spin detector

The low-energy diffuse-scattering electron spin analyzer, developed^ in the Electron Physics Group at NIST

in 1986, has led to significant advancements in the application of polarized electron studies to a large number of fields, including electron microscopy^ and surface photoemission.^ The high detection efficiency of the analyzer (I/Iq ~ 0.01), combined with a spin-analyzing

power (effective Sherman function 5eff) of ~ 0.1 leads to a figure of merit = S^^I/Iq of about 1 x 10"''. This

figure of merit is comparable with that of the best Mott

detectors. The small size of the detector (about 10~^m^)

makes this detector an extremely useful tool for a wide

range of applications. Figure la shows a sketch of the

original detector design. The principle of operation is in essence the same as a conventional Mott detector—the

polarized electron beam to be analyzed is incident upon

a gold target, and backscattered electrons are collected with two detectors in a plane perpendicular to the component of the polarization to be measured. Two sets of detectors allow measurement of the two transverse components of the spin.

The major difference between this detector and a conventional Mott polarimeter is the abihty to operate

at a low energy, i.e., 150 eV, instead of 100 keV. In order

to make low energy operation possible, the surface of the

gold target must be kept clean. This is accomplished

by operating the detector in relatively good vacuum (~

10~^ Torr), and periodically evaporating a fresh gold

film on the target. The high detection efficiency of the

detector results from the high backscattering coefficient at 150 eV, and also from the large solid angles collected by the channelplate detectors. Though the original design of the detector operates

quite well, there are two areas in which possibiUties for improvement have become evident. The improvements

have been incorporated in a new design, which is now in operation.'*

The first improvement has to do with instrumental

asymmetries resulting from the sensitivity of the detector

to angular and positional displacements of the incident electron beam. This problem can be particularly trou- blesome when the detector is used in conjunction with

scanning electron microscopy, but is of concern in other

applications as well, especially if the source of polarized

electrons is not spatially stable. The cause of the sensitivity lies in the angular dependence of the cross section
103

Incident Electrons

Incident Electrons

Input Optics

Evaporator

Target Au Evaporator

Figure 1. Low-energy diffuse-scattering electron spin analyzer, (a) Original design^ (Gl = shield grid, G2 = retarding grid) (b) New

design,* showing negatively biased electrode (El).

and the variation of the soUd angles subtended by the two electron detectors as the beam is displaced.

It was found that, by altering the geometry of the

detector and installing suitable optics at the input, in- strumental asymmetries could be dramatically reduced.

The reduction occurs because the angular dependence

of the cross section and the solid angle variation can be

made to cancel each other. ^ For example, if a paraxial incident beam passes through a simple einzel lens with focus located between the lens and the target, a

spatial displacement of the incident beam is converted

at the target into a spatial displacement in the opposite direction plus an angular displacement. The spatial

displax:ement will increase the intensity in the near detector, while reducing it in the far one because of solid angle effects. The angular displacement, however, will decrease the intensity in the near detector and increase

it in the far one because the cross section decreases for angles farther from the backscattered direction. By cor- rectly choosing the geometry of the detector, these two

effects can be made quite linear over a large range, and

so compensation involves a rather simple electron optical

design process. In a detector constructed on this principle, using a simple einzel lens, we were able to obtain

instrumental asymmetries as low a 0.0035/mm over a

displacement of up to 4 mm.

The second improvement concerns optimization of

the effective Sherman function of the detector. This was

done by careful consideration of the two major influences on the effective Sherman function, i.e., the range of scat- tering angles over which the channelplate detectors in- tegrate, and the range of inelastic electrons allowed to reach the channelplate detectors.

For 150 eV elastic scattering from an amorphous

gold target, the Sherman function generally decreases

as one goes closer to scattering in the backward direction. At 180° (full backward direction) it is identically zero by symmetry. Thus the effective Sherman function can be optimized by collecting over a selected interval of backscattering angles, usually not exceeding 150°.

In addition to its angular dependence, the Sherman

function has a dependence on the amount of energy lost in scattering from the target. Elastically scattered elec- trons have the largest Sherman function, while inelastics have a smaller and smaller Sherman function as the energy loss increases. Thus the inclusion of inelasti- cally scattered electrons in the set of detected electrons,

though desirable for increasing I/Iq, tends to decrease

the effective Sherman function of the detector. The in- elastic window must therefore be carefully controlled to balance the increased collection efficiency against the de- crease in 5eff. To accomplish this, the detector is fitted with a retarding grid to optimize the energy range of the

inelastic electrons collected. By examining the original design of the detector

(Figure la) it is evident that the retarding grid is not

optimally designed for either the scattering angle range

or the inelastic window. A suboptimal angular range is emphasized, since electrons travelling in the backward

direction pass through the grid, while those travelling

at smaller scattering angles (closer to 90°) Eire repelled

because their velocity component perpendicular to the

grid is reduced by cos 6. Also, because of this angular

dependence of the retarding energy, the cut-off energy

for inelastically scattered electrons is a strong function

of scattering angle, making it impossible to choose a sin-
104

gle inelastic window for all angles.

These difficulties have been remedied in the new design, shown in Figure lb, by moving the scattering tar- get further from the channelplate detectors and adding

a negatively biased electrode around the target region.

With this arrangement, the trajectories of the scattered

electrons from a large range of angles are made per- pendicular to the retarding grid before passing through

it. Thus, all scattering angles are treated more equally

by the retarding grid, eliminating the emphasis on low- Sherman-function backscattered electrons, and also pro- viding the same energy window for all scattering angles.

The results of the implementation of the new design

features are a Sherman function of up to 0.15 and a fig- ure of merit as high as 2.3 x lO""*. This represents a

significant improvement in the performance of the low

energy diffuse-scattering electron spin detector, and will increase its utility in a broad range of fields.

3 Use of thorium in a Mott analyzer

As discussed in the preceding section, optimization

of electron spin detectors has been a concern in our group

recently. To this end, some work has also been done

looking for ways to increase the effective Sherman function of a Mott analyzer. If this can be done without a corresponding loss in the detection efficiency ///q, per- formance of the detector can be dramatically enhanced.

Since the figure of merit of a detector is proportional to the square of the effective Sherman function, increasing

5efr has a much larger potential benefit than increasing

the detection efficiency. In addition, a higher 5eflr is ad- vantageous independent of the figure of merit, in that

instrumental asymmetries become less significant compared to a larger "real" asymmetry.

Knowing that the physical process underlying the

spin sensitivity of Mott detection is the spin-orbit inter- action, it is natural to search for a target that has the

largest possible spin-orbit effect when electrons scatter from it. Since the spin-orbit interaction increases as the

target atomic number Z gets larger, one is led to look

for suitable target materials at the end of the periodic

table. Thorium is a high-Z material {Z — 90) for which

thin target discs are readily available, so we have un- dertaken a study of the effective Sherman function of a Mott detector using a thorium target.^ The results of our study are shown in Figure 2. A GaAs polarized electron source was used as a source of polarized electrons, and

the Mott detector was of a 100 keV cylindrical design

first developed at Rice University.*^ The effective Sher- man function was measured for a 0.09 mm thick thorium

target as a function of incident electron energy over the range 20-100 keV. An energy window of 25 eV was maintained with suitable retarding voltages at the detectors.

-0.6

-0 5

-0.4

,^-0.3

-0.2

-0.1

0.0

"I " " '

1

Thorium • • \ .

• I

* ^\

i

^ Gold

i20 40 60 80

Energy (keV)

1 00 1 20

Figure 2. Effective Sherman function Sgjfversus scattering energy. Crosses represent data of Gray e< o/.®

Similar measurements were done for a 1250 A gold foil for comparison with previous gold measurements* and

the new thorium results. Also done, but not displayed

here, were measurements of the dependence of 5eff on

the size of the inelastic window.

The effective Sherman function for the thorium tar- get was found to be significantly higher than it was for the gold. At 100 keV, the gold 5eff reaches a maximum

magnitude of 0.39, while the thorium 5eff is as high as

0.49. At lower energies, the thorium effective Sherman

function is as much as 30% higher. In addition, the scat- tering intensity for the thorium target was found to be

about 15% greater than for the gold target. The combination of these effects results in about a factor of 2 increase in the figure of merit when thorium is used as a

target.

Besides improving the effective Sherman function of the Mott analyzer, we were able to provide some interest- ing comparisons with theory as well. First, our incident electron beam polarization was calibrated with a measurement at 100 keV on the gold target. A value of 0.39

is generally accepted as the correct value for the Sher- man function at this energy.* This value allowed us to put our gold measurements on an absolute scale, after which they showed agreement with previous retarding

Mott measurements,* as well as with the recent theory

of Ross and Fink.^

Comparison could then be made with similar cal- culations by Ross and Fink for thorium, and excellent agreement was found at 100 keV. The measured value of the Sherman function was 0.491 ± 0.009, and the theoretical value was 0.485. Though this does not represent a fully independent absolute experimental determination

of the Sherman function (a double scattering experiment

is required for this), it does show good internal consis- tency within the theory.
105

The above mentioned contributions to the advancement of electron-spin polarimetry will hopefully increase the prospects for "complete" electron-atom scattering

experiments, in which the polarizations of the incoming

electrons, the atomic target, and the scattered electrons

are all under experimental control in a single experiment.

Given the large body of work in the past year alone on

the improvement of spin polarimetry,'*'^'^"'^^ the future

for such experiments seems bright.

4 State-selected electron scattering from

sodium at 20 eV

Recently, our ongoing research efforts to study elec- tron scattering from sodium with spin-polarized elec- trons and atoms have produced a set of elastic scattering

data at 20 eV, and a set of superelastic data at 17.9 eV

incident energy. These data represent a very extensive

(though not yet "complete" ) set of measurements which

yields a broad spectrum of specific information for comparison with theory. Before discussing the results themselves, I would like to outline some thoughts on how measurements of the type we are involved in connect with

theoretical calculations.

4.1 The T-matrix

Most state-of-the-art ab initio scattering calculations,

such as close-coupling calculations, have as their most

immediate output a T-matrix. This matrix, which sometimes takes the form of an S-matrix or /C-matrix (i.e. reactance matrix), provides the link between an ab ini- tio solution of Schrodinger's equation and the prediction

of observed scattering intensities. Once a T-matrix is calculated, scattering intensities are arrived at through

complex scattering amplitudes, which are generally ex- pressed as sums over T-matrix elements with various coefficients, such as Clebsch-Gordan coefficients, spherical

harmonics, etc. For example, consider the scattering amplitudes for excitation of the different M-sublevels in the sodium 3s-3p transition:*

/^(«,«« Y. i'-'(2L + l)"^(.';^

I, J)

L,l=L±l ^

'

X yr^'(^,^) rf'^'izs-zv) (1) The T-matrix itself is a multidimensional matrix with

vEirious subscripts corresponding to the different channels in a scattering process. Once the incident energy of

the electron is chosen, which then becomes the energy

*This expression is shown for illustrative purposes only. It may

prove useful in the upcoming discussion, but the reader is referred

to a text on electron sc^lttering^^ for a discussion of «ill the details.

of the system in the solution to Schrodinger's equation,

all the T-matrix elements corresponding to the various

channels can in principle be calculated. The elements of the T-matrix can be thought of in a

hierarchical sense in order to facilitate keeping the multidimensional nature of the matrix in mind (see Figure 3). At the top of the hierarchy are the various energy channels open to the system. These correspond to transitions that may occur in the atomic target. There is a group

of elements for each transition energetically allowed, in- cluding no transition at all (i.e., elastic scattering). For example, in sodium, we speak of the "3s-3s" channel,

the "3s-3p" channel, the "3p-4s" channel, etc. The next level of the hierarchy corresponds to the

spin channels accessible to each transition in the scat- tering system. There may be only one channel at this

level, as would be the case for scattering from a closed

shell target such as helium at low energy where spin-orbit

effects can be ignored, or there may be several. In the

case of low-energy electron scattering from a light oneelectron atom, such as hydrogen or an alkali (considered

"one-electron" to a good approximation), there are two

channels, the singlet and the triplet. These correspond

to the two possible relative orientations of the incident

electron and the target electron. In heavier atoms, where

the spin-orbit interaction becomes significant and L — S

coupling breaks down, there may be more channels. In

fact, in this case the separation between this hierarchy

level and the next one down becomes less clear. For the

present discussion, however, we shall ignore these effects. The third level of the hierarchy corresponds to a series of orbital angular momentum channels. If no angular

momentum is transferred between the electron and the

target, this level consists simply of a series of channels

corresponding to the partial waves used to describe the

scattering. Each channel describes how a partial wave

with a particular value of L scatters from the atom. The

set of partial waves is technically infinite, but in practice

is usually truncated at some large value of L.

If, however, angular momentum is transferred be- tween the electron and the target, the angular momen- tum level of the hierarchy is split into two channels,

corresponding to exactly how the angular momentum

is transferred. For example, in the case where the atom

undergoes an 5 P transition, the extra unit of an- gular momentum can come from the Lth partial wave

by changing it into either an L -|- 1 or an L — 1 wave, according to the rules of addition of angular momenta.

There is a partial wave series for each of these ways in which angular momentum can be transferred. The T-matrix, then, has energy level channels, spin

channels, and angular momentum channels. In order
106

Energy Channels 3s-^

Spin Channels singlet triplet

Angular Momentum

Channels

Figure 3. T-matrix hierarchy for electron-sodium scattering.

to generate a simple total scattering intensity, one must sum over all these channels. Clearly, total scattering

intensities do not provide a very exacting test for theoretical predictions. Very real possibilities exist for for- tuitous cancellations between the various channels, the

result being good agreement between experiment and

theory for the wrong reason. The only way to provide a good experimental test of a theoretical approach is to ex- amine all elements of the T-matrix individually, to the extent that this is possible. This can only be done in state-selected scattering experiments in which all exper- imental variables are resolved.

Let us examine what experimental techniques can be

used to probe the different hierarchies of the T-matrix.

The first level, which consists of the atomic energy level channels, is in part trivial to sort out experimentally.

One needs only an energy analyzer on the experiment,

which is quite commonplace today. This allows measure- ment of the ground state elastic channel, as well as the

various transition channels connecting the ground state to the excited states of the atom. The first level of the Tmatrix hierarchy, however, contains many more channels

than these. For a complete measurement at this level, one must also measure elastic scattering from each of

the excited states, as well as inelastic scattering between

different excited states. Measuring these cross sections

becomes much more difficult, as it usually involves some

sort of laser excitation in combination with the electron

scattering experiment. Very little experimental work has

been done on these other channels, though they rep- resent a significant portion of the T-matrix.

The spin-channel level of the T-matrix hierarchy has

received a great deal of attention lately. The spin of the

incident electron can be put under experimental control through the use of a polarized electron source, the spin of

the atomic electron can be manipulated through optica!

pumping methods, and the spin of the scattered electron can be measured with a Mott polarimeter, as discussed

in the first two sections of this paper. A large body of work has been done on investigations involving the role played by the electron spin in scattering, though few

true "complete" measurements have been done.

The angular momentum channels in the T-matrix

present somewhat of a problem for experiments. In an

ideal world, one would like to be able to mecisure the

scattering process partial wave by partial wave, examining the contribution of each individually. This is generally impossible, though it can be accomplished in a very

limited sense by measuring at very low energies where

only the first partial wave contributes, or by studying

the somewhat analogous process of multiphoton ionization, in which selection rules limit the number of partial waves that can play a role. By measuring the angular dependence of electron scattering intensities, however, one can obtain most of the partial wave information needed. An angle-resolved

intensity is, after all, essentially a "Fourier transform" of the partial wave series, and a complete measurement over the entire angular range can be "back-transformed" to extract the coefficients for each of the individual partial waves. The coefficients are generally complex, though, so some information is lost, but the amount of information

obtained is nevertheless quite substantial.

In the situation where angular momentum is trans- ferred between the electron and the target, the angular

dependence of the intensity still provides information on

the individual partial wave contributions in a "Fourier

transform" sense. Now, however, we also have the differ- ent transfer channels, e.g., L + 1 and L — 1 in a AL = 1 transition. These can be investigated by means of align-
107

ment and orientation studies. In this type of study, the

angular momentum state of the atom is either probed

after collision in a coincidence experiment, or prepared

before collision in a superelcistic scattering experiment.

The probabihties of exciting the different M-sublevels of

the excited state, along with phase relations between the corresponding amplitudes, are measured in either case. In the coincidence measurement, the Stokes parameters

of the fluorescent light are determined; in the superelastic case the atoms are optically pumped with differ- ent light polarizations. The information obtained from

these studies relates directly back to the different angular momentum transfer channels in the T-matrix, by

way of some Clebsch-Gordan algebra or perhaps some

state-multipole formalism.

It should now be clear what is involved in generating a complete set of measurements which can verify all the predictions of an ab iniiio electron scattering cal- culation. One needs energy resolution to fix the target

excitation channel, spin resolution to fix the spin state of the scattering system, angular resolution to deter- mine the various partial wave contributions, and alignment/orientation information to separate the angular

momentum transfer channels. A complete set of measurements entails measuring all possible elastic and in- elastic excitation channels, with complete spin and an- gular momentum resolution. This is certainly a massive

task, which any single laboratory or research group can

only begin to accomplish. Nevertheless, given the large

number of research efforts in this field around the world,

it can be reasonably hoped that the combined results

will represent significant progress toward this goal.

4.2 Experimental results

The foregoing discussion of the T-matrix has hopefully provided a reasonable context for our most re- cent results on electron scattering from optically pumped

sodium with spin-polarized electrons. For this particular work, we have concentrated on an electron energy of 20 eV, and have measured both elastic scattering from

the ground state and superelastic scattering from the

first excited state (3P). The superelastic results involve

measuring de-excitation of the 3P-state in order to obtain alignment and orientation information about the

time-inverse inelastic 35—3P excitation. Thus they were

obtained with an incident energy of 17.9 eV in order to make the energy of the electrons after scattering equal to 20 eV (the excitation energy of the 3P-state is 2.1 eV).

Figure 4 shows the elastic scattering results, ex- pressed in terms of a spin asymmetry as a function of

scattering angle. This quantity, which highlights the spin

dependence in the scattering and hence in the T-matrix,

is determined by measuring intensities with incident and

I I • 1 I ' I

' I I • I I '

I I I I I

'

1 . I .-i-JL-J .1.1^

30 60 90 120 150

Scattering Angle (deg.)

Figure 4. Spin asymmetry for elastic electron scattering from Na at 20 eV. Circles, experiment; solid line, theory of Oza. '®

atomic spin either parallel or antiparallel to each other. The asymmetry is given by

-xhi 'TT (2)

where Pe and Pa are the electron and atom beam polarizations, and /n and /|| are the antiparallel and parallel intensities, respectively. Calculated in this way, the

asymmetry is equivalent to a normalized diflference be- tween the singlet and triplet scattering cross sections. It can range from a value of —1/3 when the triplet channel dominates, to a value of -f-1 when singlet dominates.

The experimental results in Figure 4 show that at this energy, the scattering is largely triplet over most of the

angular range, with a slight singlet dominance at large

and small angles.

Also shown in Figure 4 is the spin asymmetry from

a 4-state close-coupling calculation of Oza.^^ The agree- ment between experiment and theory is quite good at large angles, beyond about 60°, but significant differ- ences appear at smaller angles. Below 30°, the theory

still predicts a larger triplet cross section, while the ex- periment indicates that singlet has taken over. Combined with an absolute determination of the

differential cross section for elastic scattering from the sodium ground state, our results provide a nearly complete determination of this energy channel of the Tmatrix. All that is missing is a measurement of the

relative phase between the singlet and triplet scattering

amplitudes, which must be determined by measuring the

spin of the scattered electron while still controlling the

atomic and incident electron spins.

Our superelastic results are shown in Figure 5.^'^ Since this energy channel involves a AX of 1, we must

concern ourselves not only with the spin channels, but

also with the angular momentum transfer channels in the T-matrix. Thus there is a larger set of parameters
108

1.0

0.5 -

_r 0.0

-0.5

-1.0

0.5

J-" 0.0

-0.5

-1.0

0.2 ^

"C 0.1 EI 00

<

1-0.1

in -0.2

-0.3

1

'

1 '1

'

1

'

1

'

1 1

' 1' 1 1'

1

' 1'

1 1

'

. (a) 17.9 eV .

- •

•

• •

•

• •

•

•

•

•

. 1 . 1 . 1 . 1 . 1 . 1 . * . 1 . 1 . 1 . 1 . 1 . 1 . 1

' T

'

1 1 I

' 1 I

' 1•

(

•

1 1

'

1

' 1'

1 ! •\

Triplet

Singlet

• 'W

.i.V.iTT.i.r. i I.I.I.

7 • I

' 1 •

1

'

1 1 1 1

' I I 1 1'

1

'

1 "

t .1.1.1.1.1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1

30 60 90 120 150

Scottering Angle (deg.)

Figure 5. Superelastic electron scattering from Na at 17.9 eV.

(a) L± measured with unpolarized electrons, (b) Lx sepcirated into singlet and triplet contributions, (c) Spin asymmetry, av- eraged over Mx,-excitations.

that must be measured in this case. We have followed

the tradition of discussing the extra parameters associ- ated with angular momentum transfer in terms of physical quantities describing the atomic wave function.^*

One such quantity is L±, the net angular momentum

transferred perpendicular to the scattering plane. By

optically pumping with circularly polarized laser light

incident perpendicular to the scattering plane, we can

conveniently determine Lj. in terms of the scattering in- tensities 7+ and 7_ for 0-+ and cr" polarization of the

laser: U-I. (3)

Equation 3 shows that Lj_ is a normalized difference

between intensities associated with excitation (or de- excitation) of M = ±1 sublevels, ajid hence contains

the necessary angular momentum transfer channel infor- mation. Phase difference information, which completes

the picture for these channels, is obtained from measure- ments of P[,n and J. These physical parameters result from experiments with linearly polarized light, which we

do not consider here. Figure 5a shows our angle-resolved measurements of L± for 17.9 eV superelastic scattering. The shape of the curve is similar to most measurements of Lj. in most

systems, in that it goes initially positive, crosses over

to large negative values at intermediate angles, then in- creases toward zero at large angles. This particular curve

is interesting in that it has a double positive peak at the

smaller angles. At present, we do not have a theoretical curve for comparison with our experimented results. Having separated the angular momentum transfer channels, we must now turn to the spin channels. The

singlet and triplet spin channels are still an important

part of this inelastic scattering channel, so they must

also be resolved. The resolution is expressed by separating L± into singlet and triplet versions Lj_ and Lj^,

and by determining the spin asymmetry (Equation 2)

averaged over angular momentum transfer.^^ This is ac- complished experimentally by using spin-polarized inci- dent electrons, and by virtue of the fact that the optical pumping process automatically produces a spinpolarized excited state—spin "up" with cr'*' light and

spin "down" with (r~ light. The exact expressions for Lf^, and A in terms of scattering intensities are some- what complicated, so they are not shown here (see Ref- erence 19). and both have forms similar to Lx', the complications arise mostly from corrections for in- complete electron beam polarization.

Figure 5b shows the experimental results for Z,^ and

Lj^. There does not appear to be much of a difference

in the way angular momentum is transferred in the two

spin channels, except around a scattering angle of 50°. Here we see a prominent peak in L^, but no peak in This indicates that the second peak in the L± curve of Figure 5a is due entirely to the triplet channel.

Though there is little difference in the way angular momentum is transferred in this inelastic channel,

there is quite a large difference in the cross sections for

triplet and singlet scattering. This is seen in Figure

5c, which shows the spin asymmetry. In fact, around

30° , the asymmetry almost reaches its maximal value of — 1/3, corresponding to pure triplet scattering. It is also interesting to compare Figure 5c with Figure 4, which

shows the spin asymmetry for elastic scattering. The

two curves are remarkably similar over the entire angular

range, the difference being only that the inelastic curve

is generally larger in magnitude and has a little more
109

structure. This indicates that at this particular energy, the way in which the T-matrix is partitioned into spin

channels is quite similar for the two energy channels. As was the case for elastic scattering, these superelastic experiments constitute a nearly complete determination of all the accessible experimental information on

a particular energy channel in the T-matrix. WTien combined with an absolute cross section measurement, the

absolute magnitudes of all the relevant scattering ampUtudes can be determined. What is missing from the superelastic measurements performed to date is once again

phase difference information. Some form of phase differ- ences are measurable by scattering unpolarized electrons from atoms excited with linearly polarized light, but the

singlet and triplet contributions cannot be simply ex- tracted from these. Combining spin-polarized incident

electrons with linearly polarized optical pumping of the

target is not sufficient to extract the necessary information. Measurement of the electron spin after collision,

perhaps in combination with elliptically polarized optical pumping not perpendicular to the scattering plane, '^

is necessary for a determination of all the phase differ- ences.

5 Conclusion

The elastic and superelastic scattering results pre- sented here represent a concerted effort to learn all there

is to know about electron scattering from sodium at a

fixed scattering energy of 20 eV. Though there is a great

deal of work still to be done, we can at least say that all levels of the T-matrix (see Figure 3) have been probed in a single batch of measurements. Energy channels, spin

channels and angular momentum channels have all been

investigated, mostly in a "complete" sense, with the ex- ception of phase difference information. The two major

blocks of work still to be done involve measuring the

cross sections for as many of the transitions in sodium

as possible, and measuring with spin analysis after col- lision to extract the relative phases of all the various

scattering amplitudes. The latter of these awaits signifi- cant breakthroughs in the efficiency of spin polarization

detectors. The results described in the first two sections

of this paper represent our efforts to further the field of

spin polarimetry, in the hope that the truly complete

experiment will be realizable in the near future.

6 Acknowledgements

Major contributions to the work described here have

been made by members of the Electron Physics Group

at NIST. This work is supported in part by the U.S. De- partment of Energy, Office of Energy Research, Division

of Chemical Sciences.

References

1. J. Unguris, D.T. Pierce, and R.J. Celotta, Rev. Sci. lustrum. 57, 1314 (1986).

2. D.T. Pierce, J. Unguris, and R.J. Celotta, MRS

Bulletin 13, 19 (1988).

3. D.T. Pierce, R.J. Celotta, M.H. Kelley, and J. Unguris, Nucl. Inst. Meth. A. 266, 550 (1988).

4. M.R. Scheinfein, D.T. Pierce, J. Unguris, J.J. McClelland, R.J. Celotta, and M.H. Kelley, Rev. Sci. Instrum. 60, 1 (1989).

5. A. Gellrich, K. Jost, and J. Kessler, in Electronic

and Atomic Collisions, Abstracts of Contributed

Papers, ICPEAC XV, edited by J. Geddes, H.B.

Gilbody, A.E. Kingston, C.J. Latimer, and H.R.J. Latimer (1987), page 818.

6. J.J. McClelland, M.R. Scheinfein, and D.T. Pierce, Rev. Sci. Instrum. 60, 683 (1989).

7. L.A. Hodge, T.J. Moravec, F.B. Dunning, and G.K.

Walters, Rev. Sci. Instrum. 50, 271 (1979).

8. L.G. Gray, M.W. Hart, F.B. Dunning, and G.K.

Walters, Rev. Sci. Instrum. 55, 88 (1984).

9. A. Ross and M. Fink, Phys. Rev. A 38, 6055 (1988).

10. H. Hopster and D.L. Abraham, Rev. Sci. Instrum

59, 49 (1988).

11. M. Uhrig, A. Beck, J. Goeke, F. Eschen, M. Sohn,

G.F. Hanne, K. Jost, and J. Kessler, Rev. Sci. In- strum 60, 872 (1989).

12. See, e.g., N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions, 3rd edition (Oxford University Press, Oxford, 1965).

13. H.W. Hermann. Ph. D. Thesis, Univ. of Kaiserslautern, 1979.

14. M. Zuo, T.Y. Jiang, L. Vuscovic, and B. Bederson, in Electronic and Atomic Collisions, Abstracts

of Contributed Papers, ICPEAC XVI, (1989),

page 132.

15. J. Kessler, Polarized Electrons, 2nd edition

(Springer, Berlin, 1985).

16. D.H. Oza, Phys. Rev. A 37, 2721 (1988).

17. J.J. McClelland, M.H. Kelley, and R.J. Celotta. To

be published in Phys. Rev. A, 1989.

18. N. Anderson, J.W. Gallagher, and I.V. Hertel,

Phys. Rep. 165, 1 (1988).

19. I.V. Hertel, M.H. Kelley, and J.J. McCleUand, Z. Phys. D 6, 163 (1987).
no

COHERENCES BETWEEN AUTOIONIZING STATES OF DIFFERENT

EXCITATION ENERGIES

H.G.M. Heideman

Buys Ballotlaboratorium, Rijksuniversiteit Utrecht,

P.O. Box 80 000, 3508 TA Utrecht, The Netherlands

INTRODUCTION

In the quantum dynamical description of an

electron-atom collision process the final-state wavefunction of the atom can be represented as a coherent superposition of the possible atomic

eigenstates. For a set of degenerate substates the coherences between these substates are readily observed

by perfoming either a so-called angular correlation or

a polarization correlation experiment. However, except

in the case of quantum beats, coherences between states of different excitation energies can normally not be

observed. The differences in energy between the detected particles (electrons or photons), resulting

from different excited states, usually exclude

practical observation of these coherences. Only when

the different states can decay to the same final state can interferences and hence coherences between them be

observed. Such a situation occurs, for instance, in the near-threshold excitation of autoionizing states, where

the post-collision interaction may cause the energy

distributions of the scattered and ejected electrons,

resulting from closely separated autoionizing states,

to (partly) overlap, so that coherence studies are

possible.

In this paper we present the first experiment^

where it is possible to study coherences between states

that are widely separated in energy without utilizing

shifting or broadening efftects such as those caused by

post-coUision interactions. It concerns the excitation of autoionizing states of helium by electrons in the incident-energy domain where the energies of the

scattered and ejected electrons are about equal. In

this energy region there wUl be overlappings between

scattered- and ejected-electron energies resulting from

different autoionizing states. This provides the

possibility of different autoionizing states decaying

to final states (He"*" ion -h scattered electron + ejected

electron) where the roles of the scattered and ejected electrons are interchanged, but which are

indistinguishable. We expect that in this situation the

scattered electrons from one autoionizing state will

interfere with the ejected electrons from another one

and vice versa, and thus that coherences between the two states can be studied. To illustrate this we

consider the excitation and subsequent autoionizatioB

of, for instance, the (2s^)'S and (2p^)'D autoionizing

states of helium and the direct ionization process

e^(E^+He-> He**(' S) + e3(E^) ^ He-' + e^(E^)+ej.(E?) (l.a)

eJE^) + He-> He**('D) + ej(E^) -> He++e^(E^)+e.(E^) (l.b)1

eJE^+He- He^+e^(E^)-^ej(E9) (l.c)

where e , e„ and e- denote the incident, the scattered o s J and ejected electrons with energies E^, E^ and Ej,

respectively. The labels S, D and C refer to the two

autoionizing states and the continuum, respectively. If D S now the incident energy E is chosen such that E = E. S D * J and E^ = Ej and if electrons are detected at a fixed

direction, the two final states in (la) and (lb) are

identical and we expect interferences to occur between

scattered electrons from the 'D state and ejected electrons from the 'S state and vice versa. Of course, there is also an interfering contribution due to the

direct ionization process (Ic).

EXPERIMENT

Using a conventional electron spectrometer we have

measured ejected-electron spectra in the incident-energy range where the effects are expected to occur. For most of the experiments the apparatus is operated in the so-called constant-energy loss mode.

That is, the energy E^ of the electrons that are transmitted by the analyzer is varied along with E^,

but in such a way that the energy loss E^, defined as E^-E^-E^, is kept constant. Note that this general

definition of E applies not only when scattered electrons are detected, but also when ejected electrons

are detected. In the latter case E =E -Ef, where E^

L O J J indicates the energy of ejected electrons resulting

from a particular autoionizing state a. If now the energy loss is fixed at the excitation energy of an
autoio] ionizing state a' and we vary (and E^ we wiU

detect at E^ = Ej ejected electrons from a state a together with the scattered electrons resulting from

the selected autoionizing state a'. This is illustrated schematically in figure 1, where the two sloping lines indicate the energies of the detected scattered

electrons from the and states, whereas the horizontal lines represent the energies of the ejected

electrons from these states. Note that interferences between scattered and ejected electrons from the same

state occur at the points A and D, whereas electrons from different states interfere at B and C. Figure 2 shows a typical measurement that clearly

demonstrates interferences between scattered and

ejected electrons. The spectra, which have been taken

in the constant-energy loss mode and at a detection angle of 10 with respect to the incident-beam

direction, exhibit ejected-electron structures due to 3„ .^_2.U the (2s^)^S, (2s2p)-'P, (2p")"D and (2s2p)'P

autoionizing states of helium at 57.82, 58.30, 59.90

and 60.13 eV, respectively. In the spectra (a) and (b) the energy loss was fixed at values that do not

coincide with the excitation energy of any of the autoionizing states. So the structures in these spectra

are caused by interference between ejected autoionization electrons and the direct ionization

(2p^)^D

E('S) E('D)

continuum. As expected the two spectra look quite

similar, since one would not expect the relative phases

between the different autoionizaton processes and the

direct ionization process to change much when the energy loss is changed by a few eV. In the spectra (c) and (d) the chosen energy loss coincides with the 2 1 1 excitation energy of the (2p ) D and (2s2p) P states, respectively. So in these spectra there is not only the "background" of the direct ionization, but in addition an extra "background" due to scattered electrons from

and ^P excitation, respectively. It is clear that the interference structures in the spectra (c) and (d)

differ appreciably from those in the spectra (a) and

(b). Note also the mutual differences between the spectra (c) and (d). We attribute these differences to the fact that in

c

u,a

0)

i i El (eV)

58.64

60.46

60.14 (d)

33 34 35 36

Electron energy (eV)

Fig. 1. Schematic diagram of the energies of the

scattered and ejected electrons [resulting from

excitation and subsequent decay of the (2s^)'S and

(2p^)'D autoionizing states of helium] as a function of

the incident-electron energy E .

Fig. 2. Ejected-electron spectra, taken in the constant-energy-loss mode at different values of the energy loss E^. The nominal ejected-electron energies of

the four observed autoionizing states are indicated by

arrows. The dashed horizontal lines indicate the "background" level.
112

the spectra (c) and (d) interferences occur between

scattered and ejected electrons from different autoionizing states, whereas in the spectra (a) and (b)

this is not the case. In absence of interference between the excitation of different autoionizing states

(see reaction scheme (1)), only an extra "background"

would have been added incoherently to the spectra (a) and (b) to obtain the spectra (c) and (d). The

interference patterns would still look the same as in the spectra (a) and (b), only the non-interfering

background would be larger. It is clear from figure 2

that this is not the case.

ANALYSIS OF THE EXPERIMENTS

For the analysis of our experiments we need a theory which is capable of describing the interaction of two discrete autoionizing states with each other and

with the continuum. The simpler case of the interaction between a single discrete state and the continuum has

2

been treated by Fano and gives rise to the well-known

Fano-profile for the autoionization resonances. We have

reformulated this theory so as to account for the

state-state interferences that are involved in our experiment. Based on this generalized theory we have

made a parametrization of the doubly differential (with

respect to energy and angle) cross section for the production of scattered and ejected electrons as resulting from processes of the type shown in the reaction scheme (1). The result is dV

d£dW = X { 1

-i<5. + P„e sin<5„e -i- c.c. a a

-f-P ,e sin(5„,e + c.c. a a

(2.a)

(2.b)

(2.C)

-f-P^ .e ^'^ sin<5 e ^ sinJ -e ^ + c.c.} (2.d)

a, a a a

This expression contains contributions (i) due to the

direct ionization process (assumed to be only weakly

dependent on energy), (ii) due to interference between

each of the discrete states a and a' and the continuum

(2.b and 2.c) and (iii) due to interference between the two discrete states (2.d). The phase (and likewise S ,) is defined by

cotg(5^(E) s

E^- E

i-hT

2 a

(3)

where E is the (variable) energy of the continuum

states around the discrete autoionizing state a with

energy E^ and width /" . The amplitudes Q, P , P / and

a a a P , and the phases (p^ , (p^, and q>^ „, have a clear a , a a a a,

physical meaning involving transition matrix elements

for the relevant excitation and (auto)ionization processes and their phases. Far from resonances both

sin^ and sin<5 , are zero. When only one of them is non-zero resonance structure arises and the parametrization (2) reduces to the Fano-formula . Only

when both sin<5 and sin5 , are non-zero the state-state a a interference term (2.d) contributes. This is the case when the scattered electrons from state a' are indistinguishable from the ejected electrons from state a and vice versa. A detailed account of the theory

leading to the parametrization (2) will be given in a

3 forthcoming publication .

RESULTS

We have used eq. (2) to produce fits to the

si>ectra of the type shown in figure 2. Apart from the ejected-electron spectra shown in the figures 2(c) and

2(d), with scattered electrons "in the background", we

have also measured scattered-electron spectra with

ejected electrons "in the background". To do this the

electron spectrometer must be operated in the so-called

c

o

u

-t-J o

o

El (eV)

57.00

59.10

61.00

1 , 1

1 X

1 1 1 11 1 1 1 1

^ ^ 1 1

1

1 . 33 34 35 36

Electron energy (eV)

Fig. 3. Ejected-electron spectra taken at three different energy losses not coinciding with the excitation energy

of any of the autoionizing states involved; thus sin^^, =

0. The full lines represent fits of eq. (2) to the

experimental data (see text).
113

constant-transmission energy mode, where the energy

loss E^ = E^-E|. is varied along with at constant transmission energy E^. We can now adopt the convention

that in eq. (2) the quantities labeled with a refer to

ejected electrons and that the quantities labeled with

a' refer to scattered electrons. From the

ejected-electron spectra with sin5^,=0, hence with no

scattered electrons "in the background", we can then

determine P and q> for each of the four autoionizing

states. Figure 3 shows fits of eq. (2) to three

ejected-electron spectra for different energy losses not coinciding with the excitation energy of any of the autoionizing states involved, thus with sintJ , = 0. The

fits were obtained under the restriction that the and q> values for each autoionizing state be the same

in aU three spectra of figure 3. The quality of the

fits shows that the P^ and cp^ indeed may be considered

a a energy loss independent within the measured energy

range. In the same way we can determine the P^, and (p^, for each autoionizing state from the scattered-electron spectra with sin^^ = 0, hence with no ejected electrons

"in the background". We have found that these

2

c

o

u

• *Jo

0) w

i I

E, = 59.91 eV

i i

33 34

Electron energy (eV)

Fig. 4. Ejected-electron spectra taken at an energy loss

value coinciding with the excitation energy of the

(2p^)'D state at 59.90 eV. The experimental data are the same in the spectra 4(a) and 4(b). The full line in

spectrum (a) represents a fit with eq. (2), where the

state-state interference term 2(d) is omitted, whereas

the full line in spectrum (b) represents a fit using the

complete eq. (2).

parameters are also energy (in this case transmission energy E^.) independent within the range considered. Our verification that the parameters P^ , P^,,

and q> , are energy-independent implies that the terms

a

(2.b) and (2.c) in our cross section formula (2) are known also for ejected-electron spectra with the energy

loss coinciding with the excitation energy of any of

the autoionizing states, hence with scattered electrons

"in the background" (or with sin^^, = 1). Figure 4

shows ejected-electron spectra due to the lowest four autoionizing states of helium with the energy loss tuned so as to coincide with the excitation energy of

the (2p^)^D state at 59.90 eV; so there are

scattered electrons "in the background". The

experimental data are the same in the spectra 4(a) and

4(b). The full line in spectrum (a) represents a fit with eq. (2), where the state-state interference term

(2.d) is omitted and using P^ , P^,, <p^ and <p^, values

a a a a determined as described in the previous paragraph,

whereas the full line in spectrum (b) represents a fit using the complete eq. (2) with fixed values for P , a P ,, ffl and Q , as determined above and with P , and

a a ^a a.a ^ , as adjustable parameters. Figure 5 shows a similar measurement and fits as in figure 4, but now

with the energy loss tuned to the exciation energy of

the (2s2p)^P state at 60.13 eV; so here we have ^P

scattered electrons "in the background". It is clear

2"3

'>> c

o

o

0)

i i i i

El = 60.14 eV

j

(a)

(b)

1 1 '

1 1 1 1

1 1 1 1 11 I 1 1

I

1 1 33 34 35

' '

1 36

Electron energy (eV)

Fig. 5. Same as fig. 4 except for an energy loss value

coinciding with the excitation energy of the (2s2p)'p

state at 60.13 eV.
114

from the figures 4(b) and 5(b) that eq. (2) gives an

excellent description of the observed phenomena

resulting from the interference of different autoionizing states.

CONCLUSIONS

We have for the first time observed coherences

between the electron impact excitation of states of

different excitation energies without utilizing

shifting or broadening effects, such as those caused

by, for example, post-coUision interactions. The

method is based on the occurrence of interferences between scattered and ejected electrons resulting from

the excitation of autoionizing states by electron impact. We have developed a theory which gives an

excellent description of the observed interference

effects.We emphasize that our method is not restricted to coherence studies on states which are separated by a few eV, such as in the present experiment. It can in principle also be applied to states which lie several hundreds of eV apart, such as, for instance, in inner-sheU excitation followed by Auger emission.

ACKNOWLEDGMENTS

I gratefully mention the fruitful cooperation

with Jan van den Brink, Jaap van Eck and Gerard

Nienhuis, who greatly contributed to the work

presented here. This work was performed as a part of

the research programme of the 'Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with

financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (ZWO).

REFERENCES

1. J. P. van den Brink, J. van Eck and H.G.M. Heideman,

Phys. Rev. Lett. 61, 2106 (1988).

2. U. Fano, Phys. Rev. 124, 1866 (1961).

3. J. P. van den Brink, G. Nienhuis, J. van Eck and

H.G.M. Heideman, J. Phys. B: At. Mol. Opt. Phys.,

to be published (1989).
115

STOKES PARAMETER MEASUREMENTS IN THE HEAVY RARE GASES

J.W. McConkey*, PJ.M. van der Burgf, JJ. Corr* and P. Plessist

'Department of Physics, University of Windsor, Ontario, Canada

t Department of Physics, Laurentian University, Sudbury, Ontario, Canada.

INTRODUCTION

The study of electron impact excitation of atoms

using the electron-photon coincidence technique has

elucidated many of the processes involved in these

I interactions and this work has been extensively

I reviewed and discussed (Blum and Kleinpoppeni,

McConkey,2, Hanne^, Slevin^, Andersen et al.5.6.) For the widely studied 2iP excitation of He, LS

coupling holds strictly and the spin of the continuum

I electron can be factored out oi the problem. Thus

the positive reflection symmetry, with respect to the

scattering plane, of tne atomic wavefunction is conserved during the collision. For the heavier noble

gases LS coupling is no longer strictly valid and

I electron spin may play a role in the excitation.

Consequently states having negative reflection

symmetry with respect to the scattering plane may

also be excited. In classical terms the excitation of

1 an oscillator perpendicular to the scattering plane is

' now possible.

In previous communications from this laboratory

, (Khakoo and McConkey^.s) data for the out-of-plane

i linear and circular Stokes parameters for Ne, Ar and

Kr were presented. These enabled a study of that

,| part of the problem to which positive reflection

' symmetry applies and, most significantly, the circular

polarisation measurements enabled the sign of the

angular momentum transfer in the collisions to be

unambiguously determined for the electron scat- tering angles and incident energies considered. For

incident electron energies in the range 60-80 eV and

small scattering angles ( < 30°^ angular momentum

transfer is positive and surprisingly similar for all the

targets considered (see also McConkey et al.^). Measured total polarisations of close to unity suggested that essentially full coherence of this part of

the excitation was occurring.

Very recently Plessis et al.io reported the first in

plane polarisation correlation Stokes parameter

measurements for Kr and Xe. These enabled the

so-called height parameter, p oo» of the excited-state

I charge clouds to be obtained and thus information

' on possible breakdown of reflection symmetry in the

excitation process. The present work extends the

work of these authors to cover a wider energy and

angular range and presents details of measurements

for Ne, Ar, Kr ana Xe. In addition, a discussion is given of a number of experimental effects which can

perturb the in-plane Stokes parameter measurements.

BASIC THEORY

The density matrix description of the excitation

of the heavier noble gases leads to a formulation

involving five independent parameters rather than

only three for the simpler case of He. These

parameters and their relation to experimentally

measurable angular correlation or polarisation

correlation parameters have been outlined in the

reviews mentioned above. We refer in particular to Andersen et al.6 who provide a complete listing of

formulae linking the various parameters used by

different authors. The initial set of these parameters were the o , X. > x • cos e and cos A parameters

introduced by Blum et al." and da Paixao et al.12 More recently Andersen et al.5.6 have introduced the

0 , ,y , F*i and p 00 parameters which give a more

transparent description of the excitation process in

terms of the excited-state charge cloud characteristics. These latter parameters relate in a particularly

simple way to our polarisation correlation

measurements.8

Excitation of states having negative reflection

symmetry with respect to the scattering plane can

occur when the spin ofthe incident electron is flipped

during the scattering process. Spin-flip is due to either electron exchange interaction or to spin-orbit

interaction of the scattered electron in the atomic

field. Spin flip by electron exchange for the

ns2np5(n+l)s w?i resonance states of the heavier

rare gases, which are represented by linear combinations of singlet and triplet LS-coupled eigenfunctions, can proceed through the triplet admixture

of these states. Spin flip by spin-orbit interaction is expected to be important only for the heavier atoms.^

The p 00 parameter gives the relative probability

for spin flip perpendicular to the scattering plane

and, m terms of charge cloud characteristics, it gives

the relative height of the charge cloud. Alternatively

since the excited P state radiates like a set ofmutually

orthogonal classical oscillators we may consider p 00 as giving the relative strength of the oscillator perpendicular to the scattering plane, p 00 is related to the measurable Stokes parameters by the following

equation (Andersen et sdM):

^ ^°°^4-(l-P;)(l-F4)
116

where P \ and P 4 monitor the number of coincidences with photons polarised parallel to the incident electron beam minus those polarised

perpendicular to this beam normalised to the total

number of coincidence events, as detected by photon

polarisation analysers placed above the scattering

plane {P \) and in the scattering plane {P 4) respectively. In a polarisation correlation experiment both

in-plane and out-of-plane polarisation

mejisurements are required to measure the effects

of spin flip.

Previous experimental data which yield information on spin-flip processes are available for Kr

and Xe. 13-15 Theoretical calculations of the spin-flip

probability (or equivalent parameters) tor the

Heavier rare gases are limited. First-order manybody theory calculations (FOMBT) have been car- ried out for Nei6, Ari^ and Kris.i' and in each case only spin flip due to the exchange interaction was

considered. Recently Bartschat and Madison2o,

using a distorted-wave Born approximation

(DWBA), have investigated the importance of rel- ativistic spin-dependent effects both on the

description of the Ne, Ar, Kr, Xe target states and

on the wavefunction of the continuum electron. For

the small scattering angles involved in this work

these effects were found to be very small. This is to be expected since small angle scattering represents

distant collisions with small interactions.

EXPERIMENT

The experimental set-up has been described in

earlier papersWi, therefore only the pertinent

details will be given here. A schematic diagram of

the apparatus is given in figure 1 of the paper by

Plessis et al.io Standard coincidence techniques are used to

measure the polarisation of the photons emitted

perpendicular to the scattering plane (linear and

circular polarisation) as well as in the scattering

plane. The in-plane linear polarisation analyser

consists of a single, gold-coated mirror with an angle

of incidence of 57.5 . It is situated at right angles to

the incident electron beam direction. Both photons

and inelastically scattered electrons are detected by

channel electron multipliers (CEM). Coating the

entrance cone of the photon CEM with cesium

iodide significantly enhanced its detection efficiency

above 100 nm22, where the resonance transitions of

Ar, Kr and Xe lie. Research purity (99.995%) neon, argon krypton

and xenon were used as the target gases and were

introduced into the interaction region through a

single capillary. A pressure increase of 1.5x10-7 Torr

or less above background was maintained during

data collection in order to render negligible any

radiation trapping effects. An electron gun produced the electron beam and scattered electrons

were energy selected by a hemispherical analyser.

Electron beam currents of several piA were easily

obtained with an energy resolution of approximately

600 meV (FWHM). This resolution was insufficient to resolve the [1/2] ? ns 'P and [3/2] ? ns ^

peaks

(Racah notation) in Ne and Ar. The Kr peaks were

only partially resolved. For the electron scattering

angular range studied here, the 'P peaks of Ne and

At were approximately five times the intensity of the

respective^ peaks and so for these targets the data

are predominantly due to singlet excitation even though the transitions were not resolved. In Kr the

contribution from the adjacent peak was less than

10% while in Xe the two peal« were completely

resolved.

EXPERIMENTAL EFFECTS IN MEASURED

COHERENCE PARAMETERS

The in-plane P 4 measurements can demonstrate

pronounced minima at certain 9 ^ even though p 00 (equation 1) stays relatively constant with 0^. One

reason for this is as follows. As previously mentioned, P 4 measurements correspond to the differ- ence between the number of coincidences with

photons polarised parallel to the scattering plane

(incident electron beam) and the number of coincidences with photons polarised perpendicular to

this plane, normalised to the total number of

coinadence events. When no dipole is excited

perpendicular to the scattering plane P unity.

This is expected to be the case certainly for the lighter rare gases and measurements with He, Ne and Ar

targets have confirmed that for these gases at small

scattering angles (<30°) P indeed unity within

experimental error. If, however, there is a dipole

excited perpendicular to the scattering plane, then

the measured P 4 value will depend on the charge

cloud alignment. For example, when the charge

cloud is digned at 90° to the incident K vector, the

P 4 detector faces the 'end' of the charge cloud and

the number of detected photons polarised parallel

to the scattering plane will be at a minimum. Consequently this wdll lead to a minimum in the P^

measurement at this scattering angle, assuming that
117

the intensity of the photons polarised perpendicular

to the scattering plane does not vary too greatly with

scattering angle. In fact for cases where p oo varies

slowly with scattering angle, the minima in the P 4 curves could act as an indicator for the scattering

angle(s) at which the charge cloud is aligned perpendicular to the incident electron beam (i.e. y =

i9or).

We would point out that even if 4 is nominally

unity (no spin flip) experimental conditions can arise

which can cause the actual measured value of 4 to

deviate substantially from this value. A number of

potential problem areas have been identified and

quantified as discussed below.

(1) Finite Acceptance Angle of Photon Detector

The finite angular acceptance of the photon

detectors allows them to pick up some signal from a

dipole even though it may be pointing directly

towards the centre of the detector cone. Tnis results

in a slight lowering of the measured P values. As an

example corrections io P ^ have been calculated

using theoretical P\ values2o and are a maximum

when the charge cloud is pointing towards the P 4 detector. At this scattering angle the largest cor- rection factor is less than 1% of the measured value,

(2) Finite Beam Size and/or Offset

A much more serious problem arises if the

effective scattering plane suners a rotation from the

nominal one. (We note that a closely-related problem has been discussed by Zetner et al24 in the

context of superelastic scattering of electrons by

laser-excited targets). This is illustrated in Fig. 1 which indicates that if the scattering event occurs from a point displaced vertically from the nominal

scattering centre, then a rotation of the scattering

plane about the direction of the incident electron

occurs. The rotation angle [3 is related to the vertical displacement, r, from the nominal scattering centre, the

distance R cXo the analyser entrance aperture, and

the nominal electron scattering angle, 0 ^ according

to

tanp = r/(/?cSinej. (2)

Since the excitation process results in dipoles

excited in the scatteringj)lane then clearly, if this

plane rotates, a modification to the signals observed

by the photon detectors will occur. For the situation

discussed above, the measured P 4 can be expressed

as:

To

Anolyser

Fig, 1. Diagram illustrating the rotation^ (3 , of the

scattering plane which occurs when the scattering

centre is displaced vertically.

1 +

(3:

where, for simplicity, effects due to the

acceptance angle of the photon detector have been

neglected.

We note incidentally, that if a dipole is. excited

perpendicular to the scattering plane so that 4 1, then Eq. (3) must be modified to:

1 P ' =

1 + sin^p +

I* p.

(4)

Although Eq. (3) relates to a rather specific

scattering situation, it illustrates two important

points. First, since the effect depends on sin2 p, it is only likely to be significant for small scattering angles

( <20°) where p is largest. Second, we note that an

amplifying effect occurs for P 1 < 0 , For excitation

of the heavier gases at intermediate energies this occurs for small electron scattering angles where p

could be significant and, where, in fact our measurements are being carried out. In practice, since any electron beam will have a

finite diameter, some scattering away from the

nominal scattering plane will always occur (with

consequent rotation of the effective scattering

plane). Small vertical displacements of the electron

beam can also occur, again leading to effects similar

to that discussed above.
118

(3) Finite Acceplancs^AiigkiilAimteL.

The finite acceptance angle of the electron analyser can clearly allow scattering events to be monitored, particularly at small electron scattering

angles, where the effective scattering plane is different from the nominal one. When this occurs, a modification to the measured P 4, similar to that

given by Eq. (3) will result.

It is clear form Fig, 1 that if scattering occurs from

a point vidthin the electron beam but displaced form

the z axis then the effective scattering angle vAW be

altered. Similarly, if the electron beam gets displaced from the nominal x-direction within the

scattering plane, 9 ^ will be affected. This can have

significant effects since a particular Stokes parameter, or the alignment of the charge cloud, may vary

rapidly with scattering angle. This particular

problem as it relates to forward scattering, using an

analyser with a finite acceptance angle, has been

discussed by Martus et aL23

We note that the severity of the problem introduced by the effects discussed above depends on the

direction of observation. For example in-plane (^^4)

measurements are partiailarly sensitive to beam size or offset (effect (2)) whereas out-of-plane measurements are more sensitive to effects (3) and (4).

We have carried out a variety of calculations in

which we have considered all of the effects noted

above. We have varied beam diameters and offsets

and interaction volumes and, in some cases, have

been able to reproduce experimental data points.

An example of this is given m the follov^ng section.

EEMi^ ANO msmssiON

In our laboratoiy we have now carried out complete Stokes parameter ( 1 . 2 . 3 , 4 ) measurements

on Ne, At, Kr and Xe at a number of energies and

over a range of electron scattering angles which in

some cases extends out to 50*. Some of these data

have been reported previously, ^.s.io |n this report

we have only space to highlight a few points.

Of particular interest, in view of the proceeding

section, are the P 4 and p m data, some of which

have been presented by Plessis et al.io for 60 eV

impact excitation of &5s[3/2] ? , and 50 eV impact

excitation of Xe 6s[3/2] ? for scattering angles up to 30° lliese indicated non-zero values for Poo St these

small scattering angles suggesting that some spin-fiip

was ocoirriog. As mentioned earlier, this is not

expected based on the predicted magnitude of the

exchange or spin-orbit interactions at these small

scattering angles.

An example of our present calculations is shown

in Fig. 2 which refers to excitation of the [1/2] ?5s iP

state of Kr at 60 eV incident energy. The calculated

data set are obtained using theoretical2o Stokes

parameters and are compared with an actual data

set taken with our coincidence apparatus. The

calculated data points were obtained assuming an

electron beam radius of 1mm and a beam offset in

the -z direction of the same magnitude. Rather

similar data were obtained if we assumed all the

scattering to take place from a centre displaced 1mm

in the +z direction. The similarity between the

calculated and experimental data strongly suggests

that some net rotation of the effective scattering

plane may have occurred during the experimental

data taking even though careful alignment and set up procedures were followed.

Fig. 2 Stokes parameter data for the iP state of Kr

at 60 eV impact energy, circles- experimental data;

crosses-calculated data assuming conditions discussed in text; solid line - theory^o.

We note that the calculations show a rather small

effect for Pi + whereas it is quite dramatic for F4.

In the light of this discussion we now believe that our earlier data^o may have been affected by scat- tering plane rotation problems and hence p 00 should

be considered as being effectively zero for Kr and
119

Xe targets in the angular range considered.

Fig. 3 shows P\, P4 and p 00 data for Ne at an

incident energy of 80 eV together with theoretical

data.^7^ These experimental data have not been

corrected for any of the effects discussed in the

previous section though, there is a hint of their

presence in the P4 data. In general, however, good

agreement between experiment and predictions is obtained except for the highest angle P \ data. A

similar situation was found for Ar.

p: 0

P 1.0

00 0 - -

-ao2

Fig. 3 Ne Stokes parameter data.

Our previously and more recent data taken with

Xe targets have exhibited the most pronounced

departures from expected P 4 and p 00 values. We

are not clear why the Xe data should have been more

suspect to these effects than Kr or the lighter targets.

Experiments are continuing to clarify this situation.

Figs. 4 and 5 show some 30 eV data for the [3/2] ? 6s 3p state of Xe. They illustrate the very good

agreement between the theoretical predictions of

Bartschat and Madison^o and our data for the y and

P 3 (angular momentum transfer) parameters under

these scattering conditions. At this energy we find

that our * and P* parameters fall consistently

30-

r 0

Xe P, Eo • 30 .V

•••4

i

•Ml •

5%

•D —53

(degrees) 4?r 50 60

Fig. 4 Variation of charge cloud alignment angle,

Y , with 9 e for 30 eV impact on Xe.

some 20% below the theoretical predictions perhaps

suggesting some hyperfine relaxation caused by use

of a natural Xe mixture with a significant I 5^ 0

content.

ACKNOWLEDGEMENTS

The financial assistance of the Natural Sciences

and Engineering Research Council of Canada.

NATO Division of Scientific Affairs (RG686/84)

and the Killam Foundation is gratefully recogmsed.

0 -

RT -

-0.5-

-i.O.

Xe E„=30eV

1.

20 30

?e (degrees)

40

Fig. 5 Variation of circular polarization i-L^_) with

9 e for 30 eV impact on Xe.
120

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1. K. Blum and H. Kleinpoppen, Phys. Rep. 52*

203-61 (1979).

2. J.W. McConkey, Symp. on Electron-Molecule

Collisions (Tokyo: University of Tokyo Press)

Invited papers pp. 163-78 )1079).

3. G. Hanne, Phys. Rep. 91, 96-165 (1983).

4. J. Slevin, Rep. Prog. Phys. 47, 461-512 (1984).

5. N. Andersen, J.W. Gallagher and LV. Hertel,

Proc. 14th Int. Conf. on the Physics ofElectronic

and Atomic Collisions (Palo Alto) ed. D.C.

Lorents, W.E. Meyerhof and J.R. Peterson

(Amsterdam: North-Holland) Invited papers

pp. 57-76(1986).

6. _F/2y5.i?ep. 1^,1 (1988).

7. M.A. Khakoo and J.W. McConkey, Phys. Rev.

Lett, a, 679-82(1986).

8. _/. Phys. B: At. MoL Phys. 2Q, 5541-56 (1987).

9. J.W. McConkey, P. Hammond and M.A. Khakoo, Electronic and Atomic Collisions, ed. H.B.

Gilbody, W.R. Newell, F.H. Read and AC.H.

Smith (Amsterdam: Elsevier) pp. 105-16(1988).

10. P. Plessis, M.A. Khakoo, P. Hammond and J.W.

McConkey, /. Phys. B: At, MoL Opt. Phys. 21,

L483-8(1988).

U.K. Blum, F.J. da Paixao and G. Csanak, J. Phys.

B: At. MoL Phys. H, L257-61 (1980).

12. F.J. da Paixao, NT. Padial and G. Csanak, Phys.

Rev.A^, 1697-713 (1984).

13. 1. McGregor, D. Nils, R. Hippler, N.A, Malek,

J.F. Williams, A.A. Zaidi and H. Kleinpoppen,

/. Phys. B:At. MoL Phys. H, L411-4 (1982).

14. A. Danjo, T. Koike, K. Sani, H. Sugahara, A.

Takahashi and H. Nishimura,/. Phys. B:At. MoL

Phys.m, L595-600(1985).

15. S.J. King, P.A. Neill and A Crowe,/. Phys. B: At. MoL Phys. m, L589-94 (1985).

16. L.E. Machado, E. P. Leal and G. Csanak,/. Phys.

B: At. MoL Phys. Ji, 1773-84 (1982).

17. FJ. da Paixao, N.T. Padial and G. Csanak, Phys.

Rev.A^, 1697-713 (1984).

18. G.D. Meneses, F.J. da Paixao and N.T. Padial,

Phys. Rev. A 12, 156-65 (1985).

19. G.D. Meneses, FJ. da Paixao and N.T. Padial,

Phys. Rev. A M, 675 (1986).

20. K. Bartschat and D.H, Madison, /. Phys. B: At. MoL Phys. m, 5839-63 (1987).

21. K. Becker, H.W. Dassen, J.W. McConkey, /

Phys. B:At. MoL Phys. 11, 2535-48 (1984).

22. M.C. Johnson, Rev. ScL Instrum, 4Q, 3 1 1-5 ( 1969).

23. ICE. Martus, IC Becker and D.H. Madison, Phys.

Rev.A^, 4876(1988).

24. P.W. Zetner, S. Trajmar, G. Csanak and R.E.H.

Clark, Phys. Rev. A, In press (1989).
121

SPIN-POLARIZED ELECTRON IMPACT IONIZATION OF ATOMS

G. Baum, P. Freienstein, L. Frost, B. Granitza, W. Raith,

and H. Steidl

Fakultat fur Physik, Universitat Bielefeld, D-4800 Bielefeld 1

Federal Republic of Germany

In this report we describe the experimental

activities at the University of Bielefeld in

the area of ionizing collisions of polarized

electrons and polarized atoms and discuss results which have been recently obtained. One

apparatus can examine the spin dependence of

total ionization cross sections, and another

apparatus is capable of investigating ionization

events with scattering angles, energy partitions

and initial spin orientations resolved.

The spin dependence of total

ionization cross sections has in

the past been studied experimentally by different groups for

the following atoms: hydrogen^ " ^ , lithium''-', sodium'-'^,

and potassium' - ^ . The results

for these atoms are shown in

Fig. 1. The spin asymmetry in

ionization, Ai , is connected to

the singlet (os ) and triplet

(ot ) parts of the total

ionization cross section o for

unpolarized particles

o = {l/4)as + (3/4)ot , by

Ai = {os-ot)/(os + 3ot ) . All experiments used a crossedbeam arrangement to determine

Ai . The polarization of the

electron beam, Pe , and the polarization of the atomic beam, Pa , were measured and the observed

ion rates, Nti and Ntt, for

antiparallel and parallel orientation of the two beam polarizations,

respectively, were taken as measures of the spin dependent parts

of the total ionization cross sections. It follows that

1 Nti - Ntt

Ai = . Pe Pa Nt 4 + Nt t

As can be seen from Fig.l the

asymmetries for the light atoms

H, Li, and- Na are very much alike.

They are positive throughout the

energy range studied and have a maximum of about 0.5, indicating

that the singlet ionization cross

sections are dominating the triplet ones. For K the asymmetry is

substantial different from that of

the other atoms in the region from

threshold, Ei , to about 4Ei . The

reason for the reduced values is

not yet known; speculations involve

spin-orbit coupling effects or
122

< 0 4

>-

LU

^ 02

0.0 ' I I I I I I I llllillll

1 2 5 10 15 20 INCIDENT ENERGY ,£/ I Figure 1. Energy dependence of

the spin asymmetry Ai , measured

for hydrogen^ and the light

alkali atoms. ^ (E in units of

threshold energy)

atomic structure effects because

of the unfilled 3d subshell.

The threshold value of the

asymmetry has attracted attention. The early experiment on

could not confirm a prediction^ that should lead to Ai=l.

The experiment on Li** then conclusively showed that singlet

and triplet ionization cross

section have the same energy

dependence close to threshold.

This also implied that, right

at threshold, two-electron states

with orbital angular momenta

L>0 have to be present. This

new insight gave rise to several

theoretical treatments of spin,

orbital angular momentum, and

parity behavior near threshold. ^^-^^ Oscillations of the

asymmetry were predicted^ ^ but

could not be found experimentally

An ab initio theoretical treatment of the ionization process

at low energies from Ei to about

10 Ei is a formidable problem.

Progress in this field can only

be made by extensive interaction

between theory and experiment.

Progress has been obtained at high

and intermediate collision energies

(E > 10 Ei ) where for hydrogen

the dynamics of the unpolarized

electron impact ionization is now

well understood . 1 At low impact

energies and near threshold measurements of the triple differential cross section have been made

with unpolarized electrons for

helium^ 1 and hydrogen. The

unpolarized triple differential

measurements on He ground-state

atoms^ were analyzed for their

singlet and triplet partial wave

content up to L=2 waves. ^"^'^^ The

analysis is based on models of

threshold behavior and yields

information on triplet contributions for the special geometric

configuration of oppositely escaping electrons. Triple differential

experiments with polarized particles could provide much more insight. No results exist yet. Recent

close-coupling calculations^ o have

been predicting spin asymmetries

for angle-integrated double-differential ionization cross sections.

Measurements are needed to aid

further development.

A schematic diagram of our

experimental setup for measuring

spin asymmetries of total ionization cross sections is shown in

Fig. 2. The polarized electron beam

is produced by photoemission from

a (100) surface of a GaAsP crystal

using circularly polarized light

of a 30 mW He-Ne laser. The setup

for the electron beam and the
123

Atomic

Source Polarizing Device

Atomic Beam

Electron Beam

632 nm

Electron Polarimeter

Atom

Detection

Ion Detection

Photoemission Source Figure 2. Schematic diagram of the apparatus to measure the asymmetry

in total ionization Ai

.

electron polarimeter is very

similar to one described previously. 21 The beam polarization

is reversed repeatedly during

data taking. The electron beam

polarization is in the range of

Pe=0.35 to 0.40 and the electron beam intensity in the range

of 1=0.3 to 3 mA.

We recently studied the spinpolarized impact ionization of

He(23S) which, near threshold,

is related to that of He(l^S),

provided that the same excess

energy is available in the twoelectron escape process. For

investigating possible relativistic effects in impact ionization

and, perhaps, gaining insight

into the potassium "anomaly"

(Fig.l) we are working on an

experiment with Cs . In Fig. 2,

spin flipper22, collision region,

and Stern-Gerlach magnet for

analyzing the beam polarization

are common elements of the atomic

beam line.

For the He{23S) investigation

we use a discharge source for

producing the metas table beam.

As polarizing device a permanent

sextupole magnet with a central

stop at its exit is inserted in

the beam line. Because the helium

atom has no hyperfine structure

this magnet is able to produce

an atomic polarization of 0.9 by

focussing the ms =+1 substate atoms,

by rejection the ms =-1 atoms, and

by preferentially blocking the

ms =0 substate atoms with a central stop. The apparatus used to

produce the spin-polarized He(23S)

beam has recently been described

in detail. 2 3 Rydberg atoms in the

metastable beam interfered with

the ionization measurements. They

were eliminated V7ith the help of

a strong electric field located

prior to the collision region.

The cesium beam is obtained

from an effusive oven arrangement

which recycles the alkali metal

condensed on the first collim.ator.

With a moderate density in the

scattering region (4 x 10^ cm- ^

)

the beam can be kept continuously

running for at least one week with

a Cs filling of 5 g. Because of

the large nuclear spin value

(1=7/2) and the strong hyperfine
124

interaction of the Cs atom, a sextupole can only yield a small

spin polarization. To produce a highly spin-polarized beam we

use optical pumping, following

and considerably modifying recent

experimental work.^^ Two laser

diodes tuned to the 6^ Si / 2 (F=3)

62P3/2 (F=4) and 62Si/2(F=4)

-» 62 P3 / 2 (F=5) transitions,

respectively, are used to pump

both hyperfine sublevels of the

ground state simultaneously.

Because the light is circularly

polarized by a quarter-wave plate

only transitions with Amp =+1

are induced. After several pumping processes an atom is left

in the 5^ Si / 2 (F=4, mF=+4) ground

state. Simply reversing the sense

of circular polarization of the

pumping light yields the opposite

atomic spin polarization. An

analysis of the population distribution in the optically pumped

beam is carried out by splitting

it into the mj =+1/2, -1/2 components in a S tern-Ger lach magnet

(see Fig. 3). We find nearly all

atoms in one Zeeman sublevel, implying an atomic polarization of

Pa =0.95. The polarization direction can also be efficiently

reversed with the spin flipper.

To guide the atomic spins,

a magnetic field of 10" ^T, collinear with the atomic beam, is

present in the collision region.

The ions that are produced are

extracted with a small electric

field and directed towards a channel electron multiplier for

detection. In the case of the

measurements on He ( 223) a mass

analyzer, set to transmit He+

ions, has to be employed in order

to reduce the large background

of ions originating from. Penning

ionization of residual gas atoms

at a pressure of 5 '10"^ Torr.

Detector Position (mm)

Figure 3. S tern-Ger lach profiles

of the polarized Cs beam for different optical pumping conditions

and different modes of the spin

flipper, (a) S tern-Gerlach not

activated; (b) and (c) no optical

pumping, no spin flip and spin

flip; (d) and (e) o* pumping,

no spin flip and spin flip; (f)

and (g) o" pumping, no spin flip and

spin flip, respectively.

Background measurements are made,

in the case of Ee{2^S) by dumping

the electron beam in an upstream

section of the beam pipe connecting

the electron-source and collision

chambers, or in the case of Cs , by

blocking the atomic beam with a beam flag located 15cm in front

of the interaction region. The
125

Figure 4. (a) Arrangement of the scattering geometry with the decelerating electron-beam optics and the two spectrometers, whose axis or

rotation is coaxial with the lithium beam (not shown) . (b) Depiction

of a scattering distribution, scaled by momentum from He, with A and

B indicating the angular acceptance of the "fast" and "slow" spectrometer, respectively.

energy scale is calibrated by

observing the onset of ionization, Ei=4.8 eV for He(23S) and

3.9 eV for Cs

The apparatus to measure

the spin asymmetry in triple

differential electron impact

ionization of polarized atoms,

Ae , 2 e , also uses a crossed beam

arrangement like the one shown

schematically in Fig. 2. Studies

are made on Li atoms. The atomic

beam obtains its polarization

through state selection in a permanent sextupole magnet.

The polarization is limited here

to Pa(^Li)=0.3. All the electron

energies and momenta in an ionizing event are determined using

a coplanar scattering geometry

with two hemispherical electron

spectrometers. The coincidence

timing resolution, measured

during tests on helium, is about

15ns. The spectrometers can be

rotated from ±90° to ±150°, keeping

a separation of > 30°, and Fig.

4a shows their orientation to the

crossed beams. The acceptance

angles for the "fast" and "slow"

electrons are ±3.5° and ±10°,

respectively, (see Fig. 4b) and

the energy resolution is set to

about leV for maximum transmission.

The first major investigation will

be to measure the spin asymmetry

in the "forward" and "recoil"

peaks, near the maximum in the

total ionization cross section.

Results for the spin asymmetry,

Ai , on He (2^3) have been obtained

recently^ 5 and are shown in Fig.

5. The asymmetry is positive

throughout the energy range investigated, indicating dominance of

singlet over triplet processes.

The clear decrease of the asymme- , try and therefore increase of
126

triplet processes in going towards threshold is interesting

as it implies an increasing contribution from partial waves with

L>0. According to the analyses^ °"

1 2 of the structure of two electron wave functions at threshold, the following waves should

be strongly presented: ^ Se , ^ Po ,

1 De , ^ Fo , etc. These states are

all expected to obey the same

threshold law. Other states are

suppressed in the threshold

region. Therefore the growing

importance of triplet states

cannot be seen as a feature of

the two-electron escape process.

As all contributing states follow

the same threshold law the asymmetry should be energy independent. In order to explain the

observed decrease of Ai one has

to assum.e an energy dependence

of the initial conditions prepared in the inner reaction zone.

This provides dynamical information on the ionization process.

There is only one calculation

for comparison with the data

(Fig. 5). Bartschat ^6 used a distorted-wave Born-approximation (DWBA) , assuming maximum

interference, and included a local exchange potential. The

discrepancy in the threshold

region clearly indicates that

interaction effects, for instance

electron correlations in the exit

channels, have to be treated more

thoroughly. Qualitatively, the

slope of Ai near threshold is

also exhibited by the theory. A

comparison with the experimental values for the lithium atoms^

is also shown. Throughout the

measured energy range , the Li

asymmetry values are higher by

0.05 to 0.07, but within the

quoted normalization uncertainty

of ±0.05 the two data set could

still be considered to be equal.

For the Li data there is an indication of a decrease of Ai towards

threshold. The result of the DWBA

calculation for Li is very much

like the one for He(2='S).

0 6r -1 1 1— —i —I

Incident Energy ( eV )

Figure 5. The energy dependence

of the ionization asymmetry Ai

for metastable He in the 2^ s state

(He* ) . The full data points are

the recent experimental results^s,

the open circles are measurements

for the Li atom^ . The full curve

is the theoretical result for He* , the broken curve is for Li. Both

results are from the DWBA treatment

of Bartschat. 26

We would like to thank Dr.

K. Bartschat for communicating

results prior to publication. This

work was supported by the Sonderf orschungsbereich 216 of the Deutsche Forschungsgemeinschaf t
127

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J. Phys. E 16, 52 (1983) . G. Baum, W. Raith, and H.

Steidl, Z. Phys.D 10, 171

(1988) . R,N. Watts and C.E. Wiemann, , Opt. Commun. 57_, 45 (1986).

G. Baum, M. Fink, W. Raith,

H. Steidl, and J. Taborski,

Bielefeld Preprint 1989, submitted for publication.

K. Bartschat, Physics Reports,

in press (1989), and private

communication (1989)

.
128

"SPIN-ORBIT" EFFECTS IN INELASTIC ELECTRON-ATOM COLLISIONS: ARE EXPERIMENTS IN

CONFLICT WITH THEORETICAL MODELS?

G.F. Hanne, University of Muenster, West Germany

1. INTRODUCTION

In the past decade numerous investigations that

benefitted from progress in numerical calculations and

experimental techniques have considerably promoted

our quantitative understanding of spin dependent

electron-atom collisions. For calculations of inelastic

collisions the distorted -wave Born approximation^"^

and the R-Matrix method^ have been improved to include effects from spin-orbit interactions, and first- order many body calculations^ have been performed

which include intermediate-coupling schemes for excited states of noble gas targets. Experimentally, elec- tron-photon coincidence measurements '''^ and experiments using polarized electrons '^2 have been performed — even simultaneously^^ — to probe the electronatom collision process in greater detail than is possible

through conventional cross section measurements.

Parameters which are particularly sensitive to spin

effects have been studied thoroughly to gain models

that are adequate to describe spin-dependent colli- sions.Even though the quantitative agreement between

experimental and theoretical results is by no means

perfect — in particular at electron energies between the

excitation thresholds and 50 eV — we seem to understand quite well the spin phenomena that result from

exchange collisions and the spin-orbit interactions. There are, however, some experimental findings — "target reflection -symmetry-breaking" collisions'^ with exorbitant large relative spin-flip cross sections

for large impact parameters — which are not supported

by numerical calculations. These discrepancies are so

large, that it seems to be a qualitative rather than a

quantitative problem. This will be outlined in the

following.

den transitions will require much closer exchange

collisions, where the incident electron must deeply

interact with the electronic cloud of the target atom. Exchange collisions are therefore more probable at small impact parameters and the resultant differential cross sections show a nearly isotropic shape (see, e.g., reference 5, figures 2—6). Moreover, if the velocity of the incident electron becomes much larger than the mean velocity of target electron with which it is ex- changed, the exchange cross section becomes small and

decreases — with increasing energy — much faster than

the cross sections for direct scattering (for results see,

e.g., references 10-12, or cf figures 4 and 8).

0.2

0.0

-0.2

-0.4 I1 1 I I 1

1

1 1 1 1 p

0.2

0.0

30 60 90 120 150

SCATTERING ANGLE (DEG)

FIGURE 1. Sherman function S, and relative spin-flip

cross section p for elastic scattering of 50 eV electrons from Xe atoms. '^'^^

2. SPIN-DEPENDENT "FORCES" AND THEIR

ACTIONS

The excitation of optically allowed transitions is, in general, dominated by direct collisions, where — in a

semiclassical picture — an electron passing by an atom

creates a fast -varying electric field that can excite the atom. This is possible even for rather distant collisions,

and is associated with large cross sections which are peaked at forward scattering angles, clearly demonstrated by experimental and theoretical results formulated

within the concept of the generalized oscillator strength. In contrast, the excitation of spin-forbidFrom numerical calculations and experimental

results one can conclude that the spin-orbit interaction of the continuum electron is a short-range force, which, compared to coulombic interactions, is generally

small for low-Z targets like He (Z = 2), and for high-Z targets like Xe (Z = 54) significant spin effects are observed only if the electron comes close to the

nucleus, and is thus scattered through angles > 30° . To illustrate this we show in figure 1 experimental and

theoretical results of the Sherman function S(^) for

elastic scattering of 50 eV electrons from Xe

atoms. '6' 1'' S(^) describes the left-right scattering

asymmetry of polarized electrons (transverse polarization P)
129

where Ni and Nr are the numbers of electrons scattered to the left and to the right through the same

angle 6, respectively. Also shown in figure 1 is the

calculated relative cross section for spin flips along the

axis of incidence,

p=a{SF)/a, (2) where 0" is the differential cross section, and (T(SF) is

its spin flip part. Sherman function S and relative spin

flip cross section p are related by

S = -2 VP(l-P) sin 7 , (3)

where 7 is the difference in the phases of the two

complex scattering amplitudes (for direct and spin flip

scattering) that describe elastic collisions from spinless

targets.^'' Equation (3) shows that, in particular for small p, the Sherman function is more sensitive to spin

effects than the differential cross section to which, in the case of figure 1, spin-flip collisions contribute less than 15% at all scattering angles. In particular at small scattering angles {9 ^ 30°) the spin-flip cross section is tiny and approaches zero at ^ = 0° . In inelastic collisions the spin-orbit interaction of the conti- nuum electron with the nucleus has an even smaller

effect because it cannot produce electronic transitions between atomic states. Hence it does not contribute to the first order of the transition amplitude.

The spin-orbit interaction of the target electrons

acts in a different way. If, like in He or Na, the fine- structure splitting is small (< 2 meV) it can be neglected during the short collision process (Percival-Seaton

hypothesis), and LS-coupling is warranted for the

entire collision system. If, however, the fine-structure

spUtting is comparable to the exchange splitting — like

in Ne (2p5 3s excitation) - LS coupling does not apply

for the atomic states. Thus the total spin of the collis- ion system is not conserved even though, in Ne, the

I FIGURE 2. Semiclassical picture of electron scattering

from atoms.

|j spin-orbit interaction of the continuum electron is still negligible. In conjunction with exchange collisions

additional spin effects are then expected — often called spin-orbit (coupling) effects — which, for heavy targets, are further modified by the spin-orbit interaction

of the continuum electron.

In the following we discuss some spin phenomena

in different scattering systems studied recently in our

laboratory.

3. LEFT-RIGHT SCATTERING ASYMMETRY

FOR POLARIZED ELECTRONS SCATTERED

SUPERELASTICALLY FROM LASER- EXCITED Na*(32Pi^,3^) ATOMS

In this collision system (Z = 11 for Na) we can assume that the spin-orbit interactions during the

collision process are negligible. ^2 Surprisingly, however,

calculations and experiments 20 show that even in such a situation a strong left-right scattering or spinup down asymmetry of polarized electrons that are scattered superelastically from unpolarized excited Na*

atoms is observed.

What is the origin of such scattering asymmetries?

Let us assume, that the excited Na* atoms are in the 32Pj/2 state. A beam of unpolarized ^Pi/2 atoms can

be thought of as consisting of 50% with M = 1/2 and

50% with M = -1/2. For atoms with M = 1/2 (M is the magnetic quantum number of the total angular

momentum for a quantization ajcis normal to the scat- tering plane), the orbital angular momentum is orien- ted mainly counterclockwise in the scattering plane,

while the atomic spin is preferentially down. The situation is just the opposite for atoms with M = —1/2.

These different orientations may cause different cross sections owing to an orbit effect (figure 2): The repulsive electron-electron Coulomb interaction may be

somewhat different for electrons passing by an atom

with M = +1/2 or M = -1/2, because for M = +1/2

continuum electron and valence electron move "parallel", and "antiparallel" for M - -1/2. For the sake of

simplicity, let us now assume, that the cross section for 3^Pl/2 — 32S1/2 deexcitation collisions from M = 1/2

atoms is much larger than that for scattering from

M = —1/2 atoms. The scattering then takes place with

target atoms that are effectively spin polarized, because for M = 1/2 atoms the atomic spin is preferentially

down (see figure 2). Consequently polarized continuum

electrons show a spin-up-down scattering asymmetry

because cross sections are, owing to exchange processes,

in general different if the spin of the continuum elec- tron and the atomic electron are parallel or antiparallel.

In a similar way the superelastic scattering from

2P3^2 atoms can be considered. Within the approximation discussed in this section we find for the spin

up-down or left-right scattering asymmetry

A(2Pi/2) = -2A(2P3,2) . (4)
130

Q.

<

>-

DC

hLU

>-<

20 40 60 80 100 120 140

SCATTERING ANGLE (DEG)

FIGURE 3. Left- right scattering asymmetry A/P for superelastic scattering of polarized 10 eV electrons from excited Na atoms. Experimental [J] ^

(32P3/2); two-state close-coupling calculation

(32? 1/2)22

Any deviation from this relation indicates a spin-orbit

effect.

Recent measurements of left- right scattering

asymmetries obtained in our laboratory 21 are shown in figure 3 together with numerical data taken from a

two-state close-coupling calculation of Moores.22

There is fair agreement between the numerical and the experimental data in figure 3. Furthermore, within the

statistical uncertainty, no deviation from equation (4) has been found experimentally, thus confirming the underlying model (in the calculation this model has

been used, i.e., equation (4) is assumed to be valid).

4. DETERMINATION OF THE LIGHT POLARIZATION FOR OPTICAL TRANSITIONS IN

CESIUM AFTER IMPACT EXCITATION BY

POLARIZED ELECTRONS

Cesium atoms which are excited by collisions with

slow polarized electrons show a strong circular polarization of the fluorescence light that is emitted in the

direction of the electron polarization vector Pe- This is the result of a transfer of spin polarization to atomic

orientation by exchange collisions, l*^' 23, 24 Such investigations have been performed recently by Nass et al^S

for longitudinally polarized electrons. . »

In the present experiments^ we have measured

linear and circular polarization components (Stokes

parameters) using transversely polarized electrons, and

consequently we have detected the photons that are emitted perpendicular to Pe, i.e. perpendiculair to the

incident electron beam direction. The circular polarization (Stokesparameter 7/2) is defined by

where I((T*) and I((T ) are the intensities of light

transmitted by polarization filters which fully transmit

photons with positive and negative helicity, respectively.

In figure 4 are shown experimental results of T}2 of the 82Si/2 — 62Pi/2 transition (A = 794 nm), and of the 62P3/2 — 62Si/2 transition (A = 852 nm), plotted versus electron energy. Strong circular polarization,

caused by exchange collisions, has been found close to the excitation thresholds (3.01 eV and 1.46 eV, respectively). Even though the excitation of the 62P3/2 state

is an optically allowed dipole transition the relative contribution of exchange behaves very similar to the

(optically forbidden) excitation of the 82Si/2 state: The

magnitude of the circular polarization decreases monotonically with increasing energy, and above 8 eV no

significant circular polarization has been found. Numerical data from Nagy et al27 for the 62P3/2 — 62Si/2

transition show fair agreement with the experimental

data as shown in figure 4.

V2/P

0.2

0.1

0.0

0.2

0.1

0.0

1 1 1

1 1 .

1 III III -I—

^} (a) : \

ii III

1 1 1 1 1 1 1 1

(b)

'. \ :

1.5 2 3 4 5 6 8

ENERGY (eV)

FIGURE 4. Circular polarization (Stokes parameter

772) of light from (a) 82Si/2 - 62Pi^2 and (b)

62P3/2 — 62Si/2 transitions in Cs excited by polarized

electrons. Experimental ^ ; R-matrix calculation

.27 The excitation thresholds are indicated by a dashed line.

With respect to the topic discussed in this paper

the linear polcirization T]i for the 62P3^2 "~ 62Si/2

transition is of special interest. This Stokesparameter is defined by_ 1(45°) - 1(135°) ~ 1(45°) + 1(135°) ' (6)

^^^

I(^-) - l{a-)

l{a') + 1(0--)

(5)

where 1(45°) and 1(135°) denote the fraction of intensity transmitted by a linear polarization filter which is inclined to the electron beam direction by 45° and

135°, respectively. As has been discussed by Bartschat

and Blurn28 a nonzero value of 771 shows directly the
131

influence of spin-orbit interactions during the scattering process. In figure 5 are shown experimental results

for 7^1. We have found that T/i/P is very small even for

inelastic collisions of electrons from cesium atoms

(Z = 55!). Bartschat, however, has pointed out that, at very low energies, a small T^i does not necessarily mean

that spin -orbit interactions and relativistic effects can

be ignored in numerical calculations. 29

A(63Po) = -2 A(63P2). (9)

0.2

0.1

0.0

-0.1

-0.2 7 9 11 13 15 17 19 21 ENERGY (eV)

FIGURE 5. Stokes parameter 7j\ for excitation of the

6^P3/2 — 62Si/2 transition in Cs by polarized electrons.

5. SPIN-EFFECTS FOR ELECTRON-IMPACT

EXCITATION OF MERCURY

For one-electron atoms like Na or Cs the atomic

spin quantum number 5 = 1/2 is a well defined

quantity. The situation is different for two-electron

systems like excited rare gas or Hg atoms. For a sp (or ps) configuration the intermediate-coupling scheme

0(iPi) = a0j^g(iPi;

0(3P2) =

0(3Pi) = Q 0LS(3Pi)

0(3Po) = (p^^i'Po)

(7a)

(7b)

(7c)

(7d)

is often used. The subscript LS denotes pure Russell- Saunder states, i.e., they are coupled from the same

6s6p singlet and triplet orbitals. The intermediate

coupling coefficients are, e.g., Oc = 0.985 and P = 0.171

for Hg (6s6p) state32, and a = 0.764 and P = 0.645 for Xe (5p56s).3

For measurements of the left-right scattering

asymmetry A, cf equation (1), the intermediate coupUng has the effect that a relation like^^

A(3Pi) = -A(3P2) (8)

does not hold, in general. This is demonstrated in figure 6 where A(63p 0,1,2) is plotted against scattering

aingle for electron-impact excitation of Hg at 12 eV.30

Whereas equation (8) is clearly violated for 6 < 60° — probably because of the singlet admixture in equation (7c) — we found an approximate relation

This indicates that the spin-orbit interaction of the continuum electron and differences in the radial parts

of the wave functions (7b) and (7d), i.e. deviations

from the intermediate coupling scheme, have only little effects.

0.5

0.0

-0.5

a.< 0.5

J

1 1 1, ,

1

'

1 _

1 , , 1 , , T - ®

, , 1 ,

UJ 0.0

>

< -0.5

0.5

6

^^^^ • © * e • •

0.0

-0.5

J

, 1

1

1

'

1

' '

1

' _ 7 6

"••••••^

• • ®

•

"

1 , , 1 , 30 60 90 120

SCATTERING ANGLE (DEG)

FIGURE 6. Left-right scattering asymmetry A/P for excitation of the 63p states of Hg by polarized elec- trons at E = 12 eV.

Whereas spin effects at large scattering angles and

low energies that have been found in various experiments ^'^"'2 are supported by numerical data, at least qualitatively, some experimental data seem to indicate that spin-orbit effects occur even at distant collisions and fairly high energies that are not supported by

numerical results. As an example, we mention here the data of Plessis at al^"^ for electron-impact excitation of Xe [3/2]° 1 at E = 50 eV and 6 = 5°. In the inter- mediate coupling scheme the [3/2]° j state corresponds

to the state in equation (7c), From experimental data

of Pi and P4, which are linear polsu'ization components

obtained in electron-photon coincidence experiments,

they determined the relative spin-flip cross section^
132

(l-fPi)(l-P4)

^°°"4-(l-Pi)(l-P4) (10)

The term spin flip is here meant with respect to an

axis normal to the scattering plane. The linear polarizations Pi and P4 are defined, similar to equation (6), by

Pl,4 =_

1(0° 1(90°)

1(0°) + 1(90°)

(11)

where Pj is for photons that are emitted perpendicular

to the scattering plane, and P4 is for photons that are emitted perpendicular to the incident electron beam,

but parallel to the scattering plane. As discussed by

Anderson et al^, P4 = 1 (and consequently poO = 0) if spin flips are negligible for a s —> p excitation. The

experimental data of Plessis et al show values of poo

up to 0.15 for ^< 30°, whereas numerical results of

Bartschat and Madison^ yield practically poo = 0 (< 5- 10-3) in this range (E = 50 eV, d< 30°). Similarly, other experimental data for electron-impact

excitation of rare gases^^- ^l. 32 disagree seriously with

theory. 3. 33 'phg relative spin-flip cross section Poo ~

0.2 obtained by Murray et al^ for electron-impact

excitation of the G^Pi state of Hg (E = 16 eV, 6<

30° ) is Hkewise much larger than predicted by

theory. 34 We were interested in further experimental data

for small -cingle scattering and therefore we measured

the linear polarization Pj and P4 for electron -impact

excitation of the 63Pi state in Hg.36 Because this state has a stronger triplet character (a = 0.985) compared

to Xe [3/2]° 1 with a - 0.764, we expect that spin flip

(by exchange) are more likely in e-Hg collisions. Moreover the spin-orbit interaction of the continuum elec- tron is stronger in Hg (Z = 80) than in Xe (Z = 54).

Hence we expect that spin-orbit (coupling) effects are stronger for excitation of Hg(63Pi) than for excitation

of Xe [3/2]° 1. The experimental values of the polarizations Pi

and P4 were corrected for depolarization by radiation

trapping and hyperfme interaction. A serious problem

were proper corrections for geometrical effects. The

opening angle of the photon analyzer and the effective angular acceptance cone of the electron detection

system (including effects from incident beam diver- gence and target size) had to be taken into account. In agreement with observations of Martus et al35 a signif- icant apparent depolarisation is caused by an opening

angle effect of the electron analyzer if Pi shows a

strong dependence on the scattering angle. Thus a

deconvolution procedure had to be included as well. Furthermore we had to correct the data for estimated

misalignments of incident and scattered electron beam

direction with respect to the photon emission angle.

We found, in particular, that at small scattering angles

^e< 15° the mean orientation of the scattering plane cannot be properly estimated which resulted in large

uncertainties of the corrected data.

After all these corrections we determined the

spin-flip cross section poo from equation (10). The

result is shown in figure 7 for E = 50 eV and ^ < 30°

A theoretical curve (DWBA calculation from Bartschat34) shows very good agreement with the experimental data. From our experiments we cannot exclude

relative spin-flip cross sections of, e.g., poo = 0.01 for ^ = 0° , which still were much larger than predicted by

theory. However, as indicated by figure 1 and figure 8

0.2

0.0

0 4 8 12 16 20 24 28 32 36

SCATTERING ANGiLE (DEG)

FIGURE 7. Relative spin-flip cross section poo for electron- impact excitation of Hg (63Pi state) at 50 eV. Experimental ,36 theory 34

(which shows no excitation of the 63Po,2 states at energy losses of 4.67 and 5.46 eV, respectively, for E =

50 eV and 0°) spin flips caused by either the

spin-orbit interaction of the continuum electron or spin exchange are very unlikely at 50 eV and small

scattering angles for the two collision systems under

discussion (Hg(63Pi) and Xe[3/2]° 1 excitation).

1

6 7 8 9

ENERGY LOSS (eV)

10

FIGURE 8. Electron-impact energy loss spectrum of Hg at E = 50 eV and ^ fs 0°

6. CONCLUSIONS

While present theoretical models underlying numerical calculations work fairly well to describe spin

effects in inelastic collisions at large scattering angles
133

and low energies^ they cannot predict relative spin-flip

cross sections as large as poo = 0.15 experimentally

obtained for small—angle inelastic scattering (E =

50 eV 9< 20°) from Xe [3/2]° i excitation.l4 Our

experimental data (Hg(6^Pi) excitation) agree well — even quantitatively — with theoretical data.^* Because theoretical data yield values of Poo that are

orders of magnitude smaller than sometimes found

experimentally one explanation would be that in these

Ccises currently used numerical procedures fail completely to describe large- impact parameter collisions and/or that some, hitherto unknown, long range spin

dependent forces axe acting in these collisions. However, it is very unlikely that other, but reasonable,

theoretical models will yield such large values of poo- Thus we feel that the explanation of such data must come from elsewhere. We found it extremely difficult to control the instrumental uncertainties, in particular

at very small scattering angles, during the entire accu- mulation time of 12 hours and longer. Thus the given

error bars may still be too optimistic. A clear-cut

direct experimental observation of spin-flip colUsions

would require an experiment of the "triple scattering"

type.^'^' 3^ Such investigations, using a different experimental method, should be performed if the dis- crepancies, that are obtained by the electron- photon

coincidence method, cannot be resolved.

Acknowledgements, The author would like to thank his coworkers in Muenster for their constructive and fruitful cooperation. This work has been supported by the

Deutsche Forschungsgemeinschaft in Sonderforschungsbereich 216 Polarisation and Coirelatioii in Atomic

Collision Complexes.

References

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514 (1973)

2 K. Bartschat, D.H. Madison and G.F. Hanne, J. Phys. B 18, 1847 (1985)

3 K. Bartschat and D.H. Madison, J. Phys. B 20,

5839 (1987)

4 K. Bartschat and P.G. Burke, Comments At. Mol.

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5 N.T. Padial, CD. Meneses, F.J. da Paixao, G.

Csanak and D.C. Cartwright, Phys. Rev. A23,

2194 (1981)

6 M. Eminyan, K.B. MacAdam, J. Slevin and H. Kleinpoppen, J. Phys. B 7, 1519 (1974)

7 J. Slevin, Rep. Prog. Phys. 47, 461 (1984)

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M.C. Standage, Phys. Rev. Lett. 4, 411 (1989)

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Dynamics (New York: Plenum 1989)

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13 J. Goeke, G.F. Hanne and J. Kessler, J. Phys. B

22, 1075 (1989)

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McConkey, J. Phys. B 21 L483 (1988)

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and Ionic Impact Phenomena Vol. 1 (Oxford:

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(1986)

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19 G.F. Hanne, Cz. Szmytkowski and M. van der

Wiel, J. Phys. B 15, L109 (1982)

20 J.J. McClelland, M.K. Kelley and R.J. Celotta,

Phys. Rev. Lett. 55, 688 (1985)

21 V. Nickich, T. Hegemann and G.F. Hanne, to be

published

22 D.L. Moores, Comput. Phys. Commun. 2, 360

(1971)

23 P.S. Farago and J.S. Wykes, J. Phys. B2, 747

(1969) 24 H. Kleinpoppen, Phys. Rev. A3, 2015 (1971)

25 P. Nass, M. Eller, N. Ludwig, E. Reichert and M.

Webersinke, Z. Phys. D 11, 71 (1989)

26 F. Eschen, G.F. Hanne, K. Jost and J. Kessler, to be published

27 0. Nagy, K. Bartschat, K. Blum, P.G. Burke and

N.S. Scott, J. Phys. B 17, L527 (1984)

28 K. Bartschat and K. Blum, Z. Phys. A304, 85

(1982)

29 K. Bartschat, J. Phys. B, to be published

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(1985)

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134

Optical Analysis of Electron Impact Induced

Excited State-Excited State Transitions in Alkali Atoms*

Bernhard J. Stumpf, Department of Physics, University of Idaho

Moscow, ID 83843

*Supported by University of Idaho Research Council, Westinghouse Corporation for Idaho Nuclear

Engineering Laboratories, and the National Science Foundation (NSF-EPSCoR).

1. INTRODUCTION

Experimental and theoretical work on inelastic atom-electron collisons ha^ thus far

concentrated on the nS-n'L' problem, particularly on nS-n'P electron impact induced transitions. As is documented by other contributions to this symposium much progress has been made in understanding

this type of transition. Differential cross sections for magnetic sublevels with separation of direct,

interference, and exchange processes and even scattering phases have been reported. An excellent

review has been given by Andersen, Gallagher and Hertel.^

On the other hand, very little work has been done on the general nL—»-n'L' problem when both

L and L' are different from zero. Differential scattering data have been obtained by Bederson and

9 3 coworkers" for Na 3P—3P, 3D. Stumpf and Gallagher have measured the angular-integrated Na 3P,

M|_=:1-H-3D cross section and estimated the 3P—+3D cross section. We are currently setting up a

laboratory devoted to optical study of excited state-excited state electron-alkali collisions. This paper

describes the experimental method, derives the theoretical formulas that relate optical observables with

collision cross sections and presents numerical results using Born and Born-Ochkur approximations.

2. METHOD

Figure 1 and 2 show the experimental method in the case of cesium: In figure 1, the 6"Si

2 • 2

^ ground state is optically excited to the 6 P3 ground state. Electron impact excitation from the 6 P3

12 ^ ... 2 state to a higher n D^, state is optically analyzed by measuring total fluorescence intensity and linear

polarization of the n' "^D^^ —»-6^P^,^ transition. As a specific example, figure 1 shows the case of

impact excitation of the 7"D3 state and optical detection of the 7'D3—»-6^Pi transition. Figure 2

2 _ 2 2

_ illustrates the principle of the experimental set-up: In a crossed beam arrangement with atom beam

travelling in x-direction, electron beam and laser beam are counterpropagating along the z-axis. Total

fluorescence intensity and linear polarization are measured in y-direction. The 6"P3 state, initial state

2 for electron impact excitation, is prepared as pure spin state, |J = |, Mj=:|}=!L — 1, 5 = ^, M^^l, Mg = }i }, by a cw circularly polarized diode laser tuned to the F = 4—»F'=:5 hyperfine component of the
135

ENERGY IN CM""'

30000

20 000-

1.77 ev

10 000-

Figure 1. Cesium energy levels,

Electron impact excitation of

2 the 7 ^3/2 ^t^te from the

laser-excited 6 ^^2,/2 ^^^^^ analyzed by optical detection

2. of the 7 transition.

^3/2 S/2

1.46 eV

ELECTRON BEAM I

Figure 2

.

Experimental arrangement,

LASER PREPARED

PURE SPIN STATE

AS INITIAL

STATE FOR ELECTRON COLLISION

MEASUREMENT OF LINEAR POLARIZATION

LASER BEAM

50

40

> 30 No

o

O

O) 20

10

3P—4F

1

1

Q

1 1

•

(3P-3D. 90*) • H' —

i/ — •

P

'

1 1

' '.

1.5 2 2 5 3

CESIUM

BEAM

MEASUREMENT OF CIRCULAR POL^RIZATI0N

Figure 3. Experimental Na 3P, 3D

electron impact cross section of reference 3 (• ) compared with theoretical

calculations of reference 4 (thin,

structured line labeled "P") and ref- erence 5 (triangles) . The thick solid

line is the convolution of a step function with the measured energy distribution. A detailed discussion is given in

reference 3.

E,h(3P-"3D)
136

resonance transiton. Measurement of the total fluorescence intensity of the n D ,-^6 P ,^ transition yields, save for cascading, the 6 P3, Mj = ^—>n D , electron impact cross section.

2 Measurements of the linear polarization probes the magnetic sublevel population of the n D , state and

therefore B'Ps, Mj—n^D M , electron impact induced collisions.

2

-J

3. NOVEL FEATURES

Two features are specific of excited state-excited state electron collisions and optical studies of

these processes:

1. Both initial and final state are imbedded in a manifold of other excited states. Rich structure in

cross sections, particularly for energies below the ionization limit can be expected as a

consequence of resonances, cusps and flux redistribution. To illustrate this we show in figure 3

3 experimental data for the Na 3P, M,_= 1 — 3D cross section. Within the energy resolution of

the experiment (0.29 eV), this cross section is a step function at threshold. Unpublished close

coupling calculations by Da Paixao^ agree well with the experimental result. Calculations by

5 • / / Moores et al. are also shown. Note that figure 3 is a summary of the present status of nL-^n L

(L, L' ^ 0) electron impact excitation cross sections in the intermediate range of principal

quantum numbers n.

2. Laser preparation as described above creates a pure quantum mechanical state, an ideal situation

for collision studies. Furthermore, the laser-prepared state |L S M|_ Mg) with M(_ = l and M^^^

is rather special as initial state for electron impact excitation: Differential cross sections

(t(LM|_ = 1—+L'M^,) and (t(LM[_ = 1 ^ L' — M^,) are not equal! (For nS<-*n'L' transitions, on the

other hand, (t(M^;) = (t( — M ,) always holds.) As we will show below, this simple asymmetry

leads to a linear polarization that depends explicitly on electron exchange. No such explicit

dependence is possible after nS^n'L' impact excitation. That measurement of the linear

polarization, a very straightforward experimental procedure, yields detailed information on exchange processes, is a particularly exciting aspect of optical studies of nL—>n'L' (L, L't^O)

collisions.

4. RELATIONSHIP BETWEEN OPTICAL OBSERVABLES AND COLLISIONAL CROSS

SECTIONS.

Consider electron impact excitation from a single state of an alkali atom |L S M|_ Mg) to a

manifold of states {|L' S M^, M^,)}. After impact excitation the manifold {| L' S M , '^^g'^-^

recouples into a manifold {|j' M ,)}. After recoupling M ,)} decays to a lower manifold

M^,,)} by photon emission. We relate light intensities for the decay {|.J'M /)}^{1J" M ,,)} to cross

sections for electron impact exci tation |L S M^ Mg) — {|L' S M ,M ,)}.
137

Basic assumptions are:

• Axial symmetry for impact excitation, that is, off-axis scattered electrons are not observed.

• Validity of the Percival-Seaton hypothesis , that is M M , and M M / are good quantum

*- L ^ S numbers for impact excitation |L S M^M^)—>|L' S M^,M f)

• Fully resolved fine structure and fully unresolved hyperfine structure in the impact excited states.

The last two assumptions are obviously equivalent to the sequence t'^qJj <C t^^ t-^, <C Tj^j.^

where ^^^jj is the coUisional interaction time, t^^ and Tj^^^ are the relaxation time for fine and

hyperfine structure, respectively and Ty is the lifetime of an impact excited j' state. '''coU^^fs ^

good assumption for light alkalis but a first approximation only for heavy alkalis like cesium;

r£g-C7"j;<CTj^£g is a good assumption for alkali D states. ,

. We then have

iz = E Q(J'M ,) • Az(j'm ,)

J M ,

Ix = E Q(j'My) Ax(j'M_^,)

j'm , J where I2 and I^^ are light intensities linearly polarized along the z and x-axis, Q(J'M^,) is the angularintegrated cross section for impact excitation of the state |J M ,) from |L S Ml Ms) and A2(.rM ,) and Av (J'M 1) are optical transition probabilities given as Az(j'M ,) = 5;A(J'M ,-j"M ,)

•-

j"

j"

We use Q(J'M ,) = J2 (L' S M , M ,|L' S j' M Q(M , M ,) M^fM^ Lb J Lb

Q'^CM, ,) + Q' (M, ,) forM^, = i s _ 1

and, for a target with Ms = ^,

Q(M , M ,) =

L S {Q^M^,) forM^, = -i

where Q^, Q', are cross sections for direct, interference and exchange excitation. We then obtain

for the linear polarization, defined as,

p^. = X 100%, : Lin I(z)-hl(x) ' .
138

^Lin

^a(M ,) {Q''(M ,) + Q'(M ,) + Q^(-M ,)}

M , L5^b(M ,){Q''(M ,) +Q'(M ,) + Q^(-M ,)}

M , ^

*- ^

l'

X 100% (1)

Analogously, the circular polarization

P . =

circ X 100%

is found to be

J]c(M^,) {Q'{M^,) + Q'(M^,)-Q^(-M^,)}

P . circ _ V

2d(M ,){Q'(M ,) +Q\M ,) + Q^(-M ,)}

M ,

X 100% (2)

The coefficients a, b, c, d as a function of M , are given in the table below for D state impact

excitation and D—P unresolved multiplet detection.

Note that is summed over — M^, in equation (1); this implies, in general, an explicit exchange

dependence of the linear polarization. Only if the initial state has L = 0 Q'^''''^(M ,) =

Q

' ' (-M ,) holds and one obtains

J]a(M ,)Q(M ,) M , ^

^

^Lin ^b(M ,)Q(M ,) X 100% for L = 0 (la)

and

P . =

circ

j;c(M ,){Q(M ,)-Q^(M ,)}

M ,

*- L J2diM ,) Q(M ,) M , ^

X 100% for L = 0

6 (la) has been treated by Percival and Seaton and (2a) by Kleinpoppen .

Table: Coefficients in equations (1) and (2) for D state excitation and D—P detection.

Ml/ -2 -1 0 + 1 + 2

a -78 111 114 3 - 150

b 174 237 238 201 150

c -126 - 39 36 99 150

d 126 63 62 99 150

(2a)
139 - 6 o O

(at - dc)o/(at -''n dt)a

1/1 c .~

cj fo

y , ~ "~, c =J - ^ o

T3 ° " TMX 0. 0

a> •0 i_

i_ Ol a. 0 i. 0. U- 01 re

4-> ss !-> c ro n3 s. 0 0

ns tyt CQ

SC ro 0 C

(U L.re

rw0 rot u- 0 a.

a. jr CO

00 0 <u

s- c0 \ c

(->

-t-> 0en

reE

sf'

C

X 0 0

Q. i Q-ro

ra a. N re C3-

S- a> s- to

"3 cn 0 0 u u 0 0 2 s. X

i" re aj

O

•I- (J

TO

Q. 3z

<xI2a

L3

'. \ \ '—i ;

_t~ c ^

ire

o<

l- T3 re c o.

re= 0

qT'o - t/l M -—.

Q_ to re

O

COU "O 1- »—

•t- re re c •-.— w) ^ s- o I i_> re : O r- Q. 113 ' 03 u 2 £ d NOIiVZiaVlOd 3 U

UO

d NOIiVZJtlVIOd !IV3NI1
140

5. RESULTS FOR nP, Ml= 1, Ms = i-^n'D ELECTRON IMPACT EXCITATION IN BORN AND

BORN-OCHKUR APPROXIMATION

Figure 4 shows calculated results for Na 3P, M|_= 1, M5 = |—SD excitation and 3D—*3P

fluorescence detection, figure 5 compares these results with calculations for Rb 5P, M| — 1, Ms = 5-^6D excitation/6D^5P detection and Cs 6P, Ml= 1, Ms = 5 —7D excitation/7D^6P

detection. The comparison between Born (no exchange) and Born-Ochkur (with exchange) data in

figure 4a shows the strong exchange dependence of linear and circular polarization at low energies,

especially threshold. The threshold value for linear polarization is between 1.49% (no exchange ) and

33.3% (maximum exchange), for circular polarization between 78.7% (maximum exchange) and 100%

(no exchange), defining maximum exchange by Q^ = |Q. Born-Ochkur gives 26.0% and 85.2% for

linear and circular polarization at threshold. Figure 5a shows that the linear polarization in Born- Ochkur approximation increases with effective quantum number of the impact excited state (Na 3D:

2.99, Cs 7D: 4.53, Rb 6D 4.68). Figure 4b shows, in Born-Ochkur approximation, linear and circular

polarization for a pseudo- threshold experiment where forward scattered electrons are detected in

coincidence with photons. Figure 5b indicates that such results are crucially n-dependent. Figure 4c

compares, in Born approximation, the partial cross section Q(3P, M|_ = l—3D) with the total cross

section Q(3P—3D). Again, strong n-dependence is shown in figure 5c.

In conclusion, we have shown how optical methods can shed some first light on the yet

unexplored physics of atomic excited state-excited state transitions upon electron impact. Theoretical

calculations in Born and Born-Ochkur approximation have been presented.

ACKNOWLEDGMENT: The support of our graduate students Uma Krishnan, Leslie Wood and Gu

Zhengang and undergraduate students Mark Dexter (now at Washington State University), Rich Alva,

Russ Lawrence and Curt Martin is gratefully acknowledged.

REFERENCES:

1. N. Andersen, J. W. Gallagher and I. V. Hertel, Phys. Rep. 165, 1 (1988).

2. M. Zuo, T. Jiang, L. Vuskovic, B. Stumpf and B. Bederson, Abstracts of contributed papers, XV

ICPEAC, Brighton, England (1987).

3. B. Stumpf and A. Gallagher, Phys. Rev. A 32, 3344 (1985).

4. F. J. Da Paixao, private communication.

5. D. L. Moores, D. W. Norcross and V. B. Sheorey, J. Phys. B 7, 371 (1974)

6. I. C. Percival and M. J. Seaton, Phil. Trans. R. Soc. London A 251, 113 (1958).

7. J. Kessler, Polarized Electrons, 2nd edition. Springer 1985.

8. H. Kleinpoppen, Phys. Rev. A 3, 2015 (1971).
141

THE STEPWISE EXCITATION COINCIDENCE TECHNIQUE

M.C. Standage, W.R. MacGilllvray, A.J. Murray & C.J. Webb

Division of Science and Technology, Griffith University, Nathan, Queensland 4111, Australia.

1.0 Introduction

In this paper a review Is presented of a new type

of electron-photon correlation technique for Investigating Inelastic electron-atom collisions which

uses a combination of conventional electron-photon

coincidence and laser excitation techniques'. Target

atoras are stepwise excited by a combination of electron

and laser excitation with photons emitted from these

atoras being detected In time delayed coincidence with

electrons inelastlcal ly scattered during the collision

process. The polarized photon correlation signals so obtained are analysed to determine the atomic collision

parameters for the electron Impact excited state. Conventional electron-photon coincidence

techniques have been used in electron-atom collision

studies over the past sixteen years following the first

coincidence experiments performed by Klelnpoppen and

col laborators *

, 3 . This field has been extensively

reviewed by Blum & Klelnpoppen 1979^, Slevln 1984^ and Andersen et al 1988^. Initial experiments made use of angular correlation measurements in the scattering plane

containing the incident and scattered electron

trajectories. However, even the simplest collision

processes, such as the 'S - 'P excitation of a light L-S coupled atom, requires the measurement of polarized

photon correlations out of the scattering plane. Where

planar symmetry breaking mechanisms exist, such as strong spin-orbit interactions in heavy atoms which can cause spin-flips, measurements of polarized photon

correlations are required both out of, and in the scattering plane to obtain a full measurement of all the atomic collision parameters associated with an S-P excitation.

The introduction of laser techniques to the atomic collisions field followed the development of tunable dye lasers in the early 19'i'O's. Techniques

developed Included superelastlc scattering', atomic beam

deflection^ and stepwise excitation techniques

(MacGilllvray & Standage 1988'). Stepwise excitation

techniques have already been used in a number of non-coincidence investigations of electron-atom

collision processes that have been reviewed by

MacGilllvray & Standage 1988. Such techniques have enabled previously Inaccessible data to be obtained on

partial total cross sections for the electron Impact

excitation of metastable states. The narrow bandwidth

available from lasers has been used in experiments In which the role of spectroscopic structure in electron-atom collision processes has been investigated.

Such methods have also been used to provide data which

is free from the depolarizing effects of atomic

hyperflne structure. Stepwise excitation techniques have

also provided a new method for investigating the electron impact excitation of VUV transitions.

One of the objectives of the research described in this paper has been to demonstrate the feasibility of using

stepwise excitation coincidence techniques and to show

their application to the study of the electron Impact

excitation of VUV transitions, which have been of such general Interest In this research field. Many ground to first excited state VUV transitions of interest In atomic collision investigations are connected to higher

lying transitions In the visible part of the spectrum so that stepwise excitation techniques can be widely

applied. Such an excitation scheme, which combines VUV

and visible transitions has already been investigated by

the authors' using non-coincidence techniques to measure

partial total cross sections for the 6'PiStato of mercury. In these experiments, the initial electron

excitation of the 6»So - 6'Pj (185nm) VUV transition was followed by laser excitation of the 6'Pi - e'Dj (579nm)

transition. Fluorescence was observed on the G'Dj - 6'Pi

(313nm) transition. The lifetime of the 6iPi state is short (1.3ns) while the S'Dz state lifetime is 11ns. Consequently, the expected duration of the coincidence

signal is of the order of 20ns. Considerable hyperflne structure Is present in these transitions due to non-zero nuclear spin isotopes of mercury. However,

because of the reduced Doppler width of the mercury

atomic beam and the use of a single-mode dye laser,

resolution of most of the spectroscopic structure of the laser excited transition was achieved so that data could

be obtained free of hyperflne structure effects. The measurement of atomic collision parameters

for the 6'Pi state of mercury Is of considerable

interest because recent measurements reported for heavy

atom targets, such as the rare gases, show a marked departure from light atom behaviour with the observation
142

of spln-flJp cross sections even at relatively small scattering angles (see reference 6 for a review of this data). This effect Is attributed to spln-orblt

interactions betweeri the target atom and the colliding

electron. Mercury is the heaviest atom so far investigated In atomic collision studies and It has a large spin-orbit interaction, so that significant

spin-flip cross sections would be anticipated.

Attention is focussed in this paper on techniques

in wlilch a complete measurement is obtained of atomic

collision parameters for electron Impact excitation of atoms by a non-polarized beam of electrons. To achieve

tliis for collisions Involving heavy atoms requires the measurement of four atomic collision parameters for a 'S

- 'P excitation, which in a conventional coincidence

experiment requires polarized photon correlation measurements to be made both perpendicular to and In the scattering plane. The laser polarization provides an additional experimental parameter which enables all atomic collision parameters to be obtained with

correlation measurements made only perpendicular to the scattering plane. 2.0 Theory

MacGlllivray & Standage 1988 discuss In detail

the theoretical form of the stepwise coincidence signal

and here only the results of such calculations are presented. A typical excitation/emission geometry is shown in figure 1 together with a typical energy level diaRram for a stepwise excitation experiment. Atoms In a

X

Laser beam

Electron beam

-|l>

Laser f\

Excltatloiy ^Fluorescence

-/-|e>

Electron / -^|f>

Excitation / —^Ig>

^^Fluorescence

y Detection

£ = Laser Polzn. Vector

£ = Detector Polzn, Axis

J = 0 ground state are first excited by electron Impact

to a J = 1 excited state and then by laser excitation to

a higher J = 2 state. Fluorescence is detected from the stepwise excited atoms as they spontaneously decay back

to a J = 1 state. A mutually orthogonal excitation/

emission geometry Is shown in which laser and electron

beams travel along the x and z directions , respectively

and the x-z plane is the scattering plane. Fluorescence

Is detected along the y direction.

The stepwise coincidence signal is expressed in terms of Stokes parameters as Pi = (lo - I^o)/I. P2 =

(1^5 - Ii35)/I. Pa = (I RHC where I is the

total Intensity and the linear polarizer angle Is given

with respect to the z direction.

For collisions which Involve heavy atoms with

strong spln-orblt interactions, four atomic collision

parameters are required to describe the electron Impact

excitation of an S - P transition. In the collision

frame, where the quantization direction is taken as parallel to the incident electron beam direction, the density matrix elements of the P state may bo parameterized as

A = poo/o. cosx = Repio/(PooPi 1 )'

sin* = Impio/(PooPi 1 ) . cos6 = Pi-i/pn (1) a = Poo +aPii

where diagonal density matrix elements represent

differential cross sections for magnetic sublevels and off-diagonal elements represent excited state coherences

produced by the collision process. a is the differential cross section. In the natural frame the quantization axis is taken perpendicular to the scattering plane and the four independent parameters

normally used are poo. the normalized spin-flip cross

section, Lj^, the angular momentum transferred to the atom

during the collision, 7, the tilt angle of the major

axis of the atomic charge cloud to the Incident electron

beam direction and P , the polarization of the charge

L cloud measured perpendicular to the scattering plane. The coherent nature of the excitation is characterized by the coherence parameter /j, which is given by the expression'"

Fig. 1 The Excitation / Emission Geometry.

Inset : Energy Level Diagram.
143

2 Pa

P2

1

/- ) (cos^x + sin2*) (2)

11 - COS®

and the vector polarization parameter |P | (P, ^4'

where |P|>|^y|>0. It should be noted that the expression

for differs from that normally used in the literature

by a factor 2/(1 - cos5) which arises when it is correctly derived using the expression given above. It is this expression that properly satisfies the inequality. Although the coherence parameter Is frame dependent, whereas |P| is not, and for this reason |P| has been favoured as the measure of coherence, the coherence parameter can provide a more stringent test of the coherent character of the collision process by

virtue of the inequality stated above. This has recently

been noted in work on the comparison of superelastic and coincidence experiments on sodium' . > Consideration has to be given to the atom-light

interaction, which if it can be regarded as a weak

interaction involving only single photon processes, then enables analytical expressions to be obtained for the stepwise coincidence signal. For the geometry and excitation/emission scheme described above, it can be shown that these signals are given by'

27 9 9

IPi = ^^~2 ' 6cos2a] + -cos2a + -(l-A)cos5

IP2 = -3(5 - cos2a) [A(l-A) ] cosx (3) IP3 = 9(3 ^ cos2a)[A{l-A)] sin*

I = A(6cos2a+li) + (5cos2a+i|) + i4{ 1 -A ) ( 8sl n^a-g ) cos6

where a is the laser polarization angle. In the event

that the atom-light interaction is strong, then a quantum electrodynamical treatment is required. Space

does not allow a detailed discussion of the QED

calculation here (for a dicussion see references 9 and

12), but results from calculations are considered In Section 3.0.

It can be seen from the above equations that the additional experimental parameter provided by the laser polarization enables more information to be obtained

than would be possible with the same geometry and a conventional coincidence experiment. The parameters A and 5 may be determined provided measurements of Pi are made for two laser polarizations such as a = 0" and 90". The remaining parameters cosx and sin* may then be

determined from measurements of P2 and P3. As mentioned

In Section 1.0. conventional coincidence measurements of these parameters require an additional linear

polarization measurement in the scattering plane. 3.0 Experiment

A schematic diagram of the apparatus is shown in figure 2.

Interference Filter

/ / \ Photomultiplicr Tube

Lens X/l plale Linear

polarizer

Fig. 2 Schematic of the Experimental Apparatus.

The apparatus is the same as for a conventional

coincidence experiment with the addition of a laser system to provide the optical excitation step. The electron gun is of a two stage aperture lens design with

a tungsten filament as the electron source. The gun was operated at 16.0 eV with a beam current of 500 nA and an energy halfwidth of 500 meV. The electron analyzer was

of a standard cylindrical design with an aperture lens fitted on the Input to accurately image the interaction

region into the analyzer. The acceptance solid angle of the analyzer was 0.01 sr. A quartz lens Imaged light from the Interaction region into an optical detection

system mounted inside the vacuum system to maximize its efficiency. The system consisted of an externally

rotatable linear or circular polarization analyzer and

an Interference filter and a cooled photomultipller

tube. The acceptance solid angle of the system was 0.50 sr.An atomic beam oven produced an effusive beam of atoms with an estimated density of (2-5) x 10'* atoms/m'
144

witlun the interaction region. Tiie dimensions of the interaction region were determined experimentally by

performing conventional electron-ptioton coincidence measurements on mercury atoms excited by direct electron

impact to the nearly degenerate manifold of singlet and

triplet 6D states with photon detection at 579 nm along

a direction perpendicular to the atomic beam. The size

of the laser beam was chosen to match the dimensions of the Interaction region.

An actively stabilized cw ring laser was used with a jitter bandwidth of about 1 MHz. Typical laser powers in the interaction region were 350 mW with the laser beam having a 1/e diameter of 2.5 mm. The 579 nm transition of mercury which was excited by the laser has extensive hyperflne structure, most of which was resolved in this experiment. The laser was tuned to the zero-nuclear-spin isotope peak which enabled data to be obtained free from the depolarizing effects of hyperflne

structure that would be present in conventional

coincidence data. Careful attention was given to possible sources systematic errors caused by stray fields and radiation

trapping, llelmholtz coils and mu-metal shielding was

used to reduce stray magnetic fields to 10-* T. The 185 nm transition of Hg has a short lifetime and a correspondingly strong absorption coefficient so that

the presence of radiation trapping has to be considered.

Wo found that it was possible to induce significant

radiation trapping at higher pressures as evidenced by

Increased decay times for stepwise coincidence signals

and accordingly operated the experiment at tank pressures that reduced such effects to the level at which a small correction of a few percent was applied to the data. It should be noted that in this experiment,

the use of stepwise excitation confines radiation

trapping to the region of the interaction volume

illuminated by the laser beam, thereby limiting this source of systematic error. Figure 3a shows a stepwise coincidence signal

obtained with the polarizer in the photon detector

removed. Standard coincidence techniques were used to obtain the signal. The electron analyzer was tuned to the 6iPi peak of the Inelastically scattered electron

energy loss spectrum. Real coincidences can only Involve

313 nm photons emitted by stepwise excited atoms which

have previously scattered an electron from the 6'Pi level that Is detected by the electron analyzer. The

total run time to accumulate this signal was 12 hours with a 6'Pi Inelastic electron countrate of 15 kHz and a photon countrate of 130 Hz. The photomultlpller tube dark countrate was 10 Hz and the countrate from direct

electron excitation of the states of Hg, which

have transitions to lower states at aproxlmately 313 nm

,

was 30 Hz.

c

Time (ns)

Fig. 3(a). The Stepwise Coincidence Signal.

0 20 40 60 80 100 Time (ns)

Fig. 3(b). The Direct Coincidence Signal.
145

Qualitatively similar coincidence signals were

obtained when a linear or circular polarization element

was placed in the photon detector. Measurements of the Stokes parameters were obtained from coincidence signals

that were normalized to the number of scattered

electrons detected during a run. Many runs of short duration were recorded and statistically analyzed to

I ninimlze systematic errors that might arise because of

' drifts In experimental parameters.

4.0 Results and Discussion

Before considering the results obtained from

I measurements of atomic collision parameters, we consider

more closely evidence that a weak excitation treatment

of the data obtained in this experiment is adequate. If this is the case, then the coincidence data may be

. analysed using the analytic expressions of equation 3. Figure 3b shows a direct coincidence signal obtained by

tuning the electron analyzer to the S^'^D inelastic

scattering peak. The expected lifetime for the direct

coincidence signal is about 11 ns, which Is what Is observed, however the striking feature of the stepwise

coincidence signal is that it has a decay time of only 5

ns. The expected decay time for this signal under weak

excitation conditions Is similar to the direct signal,

approximately 11 ns. Some results are now considered from an extensive

1 series of QED calculations we have performed to model

,| the interaction of strong laser light with the laser

excited transition. The QED model produces time resolved

population and coherence data for both the upper and

lower states of this transition. In practice, the shape

of the coincidence signal is found to be relatively

Independent of the precise values of the atomic

collision parameters, whereas the total Integrated area

of the signal does depend on their values. For the purpose of a comparison with the weak excitation case,

I arbltary initial values of the atomic collision

'; parameters of A=cos6=cosx=sin*=0 . 5 were used to

1^ calculate the curves shown in Figure 4. The laser

Ij intensity was assumed to be constant over the laser beam

I which was assumed to have a diameter of 2.5 mm and

' calculations were performed for 1 mW power to simulate

weak excitation conditions and at 1000 mW power to

I represent strong excitation conditions . The decay time

for the 1 mW signal is approximately 11 ns whereas the

1000 mW signal has a 1.5 iis decay time. Some evidence of

a Rabl nutation can be seen on the 1000 mW signal with a corresponding Rabl frequency for the S'Pi - e'Dz (579nm)

transition of ilg of 130 MHz.

0 10 20 30 40 SO

Time ( iiS )

I'ig. 4 . Time Resolved Stepwise Coincidence Iiilciisily .

The cause of this enhanced decay rate is the coupling by the laser of the upper excited state, with a lifetime of 11 ns to tlie lower excited state, with a

much shorter(1.3 ns) lifetime. When the effect of the Gaussian laser beam intensity profile is included, the result is that the

range of Rabl frequencies present removes the oscillations from the signals and Increases the decay

time. When the Doppler spread of the atomic beam is also

Included and the laser beam parameters are adjusted to match the experimental conditions used, good agreement

is obtained with the measured decay time. When the same calculation is applied to determine

the behaviour of the Stokes parameters, the variation

from the weak excitation result is between 2% and 5*, depending on which parameter is chosen. It can be seen

that in spite of the presence of factors which quite

strongly influence the decay rate of the stepwise

coincidence signal, little change occurs in the Stokes

parameters

.
146

Scancnng Angle 9, = +10° 9, = +20" 9, = +30"

Collision Frame

X 0.47 ± 0.04 0.43 ± 0.08 0.36 ± 0.06

cos 5 - 0.01 ±0.15 - 0.08 ± 0.40 -0.24 ±0.21

cos X -0.13 ±0.09 0.05 ±0.12 0.42 ± 0.25

sin <() - 0.05 ± 0.03 - 0.03 ± 0.02 - 0.08 ± 0.05

0.14 ±0.08 0.06 ±0.10 0.43 ± 0.25

p 0.34 ± 0.07 0.18 ±0.07 0.55 ±0.31

Natural Frame

Poo 0.26 ± 0.05 0.26 ±0.16 0.24 ±0.10

0.33 ± 0.07 0.18 ±0.07 0.54 ± 0.31

Y + 16" (+10" -20") - 10" (+34" -15") - 48" (+6" -31")

Lt +0.06 ± 0.03 +fl.04 ± 0.03 +0.10 ± 0.07

Table 1 The Atomic rnlli.sion Parameter.s

Table ] shows data obtained for collision frame

and natural frame parameters for three electron

scattering angles and an Incident electron energy of 16.0 ± 0.5 eV. Experimental uncertainties are one standard deviation and ± 1.5" in the electron scattering

angle. The data was obtained for negative electron

scattering angles i.e. the electron detector lay in the

(-X, 7.) quadrant of the scattering plane. Sign changes

have been applied to collision parameters, where

appropriate, to correspond to positive electron

scattering angle. The data clearly shows a breakdown of positive

reflection symmetry with respect to the scattering plane

with the spin-flip cross section Poo > 0 and cos5 = -1. The angular momentum transferred in the collision is small and the coherence and total polarization

parameters show a marked loss of coherence in the collision process. Not surprisingly, the data shows a marked departure from that expected for light-atom

collisions and, particularly with respect to spin-flip

cross sections, reflects the behaviour found in electron

- heavy rare gas collisions.

5.0 Conclusion

We have seen that the Introduction of a laser excitation step between the initial electron impact

excitation process and the final fluorescent emission

process enables the unique properties of laser light to

be utilized in coincidence experiments. The laser

pplarizatlon may be used as an additional experimental

parameter to reduce the number of measurements required

to fully determine the atomic collision parameters. The spectral resolution may enable the effect of spectroscopic structure on the measurement of atomic

collision parameters to be eliminated. The optical power

of the laser light enables a satisfactory stepwise

signal to be generated even when a very rapid decay rate exists for the electron-impact excited transition. A tightly collimated laser beam allows the interaction

volume to be accurately defined and confines radiation

trapping to this volume, thereby minimizing such effects

In the case of electron excited VUV transitions,

the main advantage of the stepwise excitation technique

lies in the transfer of polarization measurements to wavelengths at which standard optical components can be used. Finally, stepwise excitation techniques open up the possibility of applying coincidence techniques to the investigation of the electron-impact excitation of metastable states. Although the stepwise excitation technique has the obvious practical disadvantage that a laser system

Is required In addition to the usual electron scattering

apparatus, the advantages outlined above will hopefully

encourage more researcliers to consider using such techniques

References

'A.J. Murray, C.J. Webb. W.R. MacGillivray & M.C. Standage, Phys . Rev. Lett. 62, 411 (1989).

^M. Eminyan, K.B. MacAdam. J. Slevin & H. Kleinpoppen,

Phys. Rev. Lett. 31, 576 (1973).

J J phys.B. , 7, 1519 (1974) . ^K. Blum & H. Kleinpoppen, Phys. Rep. 52, 203 (1979).

sj. Slevin, Rep. Prog. Phys. 47, 461 (1984).

*N. Andersen, J.W. Gallagher & I.V. Hertel, Phys. Rep.

165, 1 (1988).

^I.V. Hertel 4 W. Stoll in " Advances in Atomic and Molecular Physics' eds D.R. Bates & B. Bederson (Acad. Press, N.Y. 1977) 13, 113.

8B. Jaduszliwer, G.F. Shen. J-L Cai & B. Bedersen, Phys.

Rev. A. 31, 1157 (1985). «W.R MacGillivray & M.C. Standage, Phys. Rep. 168, 1 (1988)

'"A.J. Murray, C.J. Webb, W.R. MacGillivray & M.C. Standage, in preparation (1989). "P.M. Farrell, C.J. Webb, W.R. MacGillivray & M.C. Standage, submitted to J. Phys.B., (1989).

'2P.M. Farrell, W.R. MacGillivray & M.C. Standage, Phys.

Rev. A.. 37. 4240 (1988).
147

SUMMARY OF SYMPOSIUM PROCEEDINGSSOME IMPRESSIONS AND OBSERVATIONS

Benjamin Bederson

Physics Department

New York University

New York, N. Y. 10003, U.S.A.

Being invited to present the

closing talk at a specialized

symposium is a mixed blessing

at best. Is one supposed to

distill the essence of the

previous talks and present a global, definitive overview

that effectively summarizes for

posterity the important

material already offered by the

previous speakers? Or should

one use the previous talks as a

springboard, to offer a

projection of what is to come

at the next such symposium?

Either of these extremes

requires a presumptuousness

that I don't particularly wish

to assume. On the other hand,

knowing that I would be giving

this talk resulted in the

obligation to listen to all the

talks presented (I noticed that

many other attendees did the

same even without the unique

incentive that drove me.)

Accordingly I feel that I can

assume the luxury of offering

remarks that convey impressions

as opposed to definitive value

judgements.

At least I have the advantage

over my late NYU colleague

Werner Brandt, who was in a similar position at a symposium

on beam-solid interactions held

in Tokyo some years ago. Werner

had simply neglected to obtain

the required visitor's visa,

and so was refused access to

the meeting, being forced to

stay during the entire meeting

at the airport. I understand

that he did finally get

clearance to attend the meeting

on its last day. He gave the

final talk as scheduled,

anyway

.

While listening to the several

talks presented at this meeting

and although they covered a

quite wide range of subject

matter, both theoretical and

experimental, it was not too

difficult to identify the

common thread that ran through

most if not all of the talks.

It is, in my opinion, the

appeal this kind of work has

to those wishing to effect as

close a connection between

fundamental theory,

computational techniques, and

experiment as is currently

possible. And despite the fact

that there was not an

overwhelming number of talks--

only about two dozen--virtually

all aspects of current activity

were covered to one degree or

another. (I should add that the

relatively leasurely pace of

the meeting allowed each talk

to be long enough to be

understandable to the nonexpert, for the most part.) In

fact in considering what can be

accomplished by the application

of the state-of-the-art

experimental techniques

discussed here one cannot help

but be impressed by the

virtually unique role this

field is playing in collision
148

physics, and this statement

applies equally well in atomic,

nuclear, and high energy

collision studies.

Before discussing some of the

specifics of the new work that

surfaced at the meeting, I

would like to address the

points singled out for special

mention by Joachim Kessler in

his introductory talk, and for

which he expressed a hope that

these would be resolved during

the meeting.

First Prof. Kessler raised the

question: should there be a conference such as this one

every two years, a strong

argument in favor being that it

is correlated and coherent with

the biennial ICPEAC. But as he

points out this is not a sufficient condition for a positive response. There has to

be an overdriving scientific

purpose that is not already

satisfied in another way (say

at the ICPEAC itself) . Based on

what happened at this meeting,

I agree with Prof. Kessler that

the answer is yes, although

some minor suggestions for

variations will be offered at

the end of this paper.

Also, Prof. Kessler mentions

the experiment of the JPL group

(Trajmar et al) on laserexcited barium that revealed an

unexpected asymmetry of

scattered electrons with

respect to a direction along

the electric vector of the

exciting laser beam, when

linearly polarized. They could

not make the assymetry

disappear despite many efforts

to discover systematic errors

by elaborate variations of

experimental paramters (in this

as with other specific papers I

refer the reader to the

original symposium articles for

reference citations) . The

conclusion of the JPL workers

at the time was that either

there remained an as yet

unrevealed systematic error or

there was a true asymmetry that

reflected some not-understood

breakdown of collision theory.

The JPL talk definitely answers

this question in favor of the

former (to most everybody's

regret, I'm sure). The problem

concerns the subtle influence

of imperfect angular alignment

on the polarization detection.

The effort put into this work

by the JPL group will likely

help others in identifying

similar systematic deviations

from nominally mututally

perpendicular crossed beams.

A related discussion on overlap

integrals and sensitivy to

orientation of the scattering

plane with respect to the

detection system was given by

Prof. McConkey (Windsor) . I

refer the reader to the

penultimate paragraph of the

section of that article

entitled "Finite beam size

and/or offset" for their

important conclusions.

A second, also "technical"

question raised by Kessler

concerned the assumed accuracy

of electron polarization

determination by Mott

scattering. Two papers at the

meeting addressed this

question, one by a University

of Missouri group (Khakoo et

al) , and the other by the

Munster group » This talk was

originally to have been given

by K. Jost but as we know he

unfortunately passed away just

before the ICPEAC, so the paper

was given by A. Gellrich. As

can be seen from both these

papers, the last word has yet
149

to be spoken on this subject.

The conservative view expressed

by the Munster group,

reinforced by the earlier

systematic investigations of

Fletcher et al, to the effect

that current Mott analyzers

generally can yield accuracies

of perhaps 5%, remains the

safest one to date. Future

improvement, particularly when

calibration to a simpler

physical situation, i.e.,

polarization of helium impact

radiation, is promising but yet

remains to be definitively

established. A third paper, by

the NIST group (McClelland) is

interesting for a different

reason since it takes an

alternate approach using low

voltages to the design of a Mott analyzer (see the

McClelland paper for details)

.

Several other issues of

specific interest or should we

say that presented potential

traps for experimentalists were

raised at the meeting. One

concerned radiation trapping

(Standage et al) —always a

possible problem in dealing

with resonant radiation--, and

another with background timedependent pressure effects

caused by mechanical beam

chopping (Lubell)

.

I found it very interesting to

note that all four of these

experimental problems have

challenged collision physicists

to one degree or another

throughout the history of

crossed-beam research

(radiation trapping: Mitchell

and Zemansky; Mott scattering:

Mott and Massey; "overlap

integral" effects in angular

distributions: Massey and

Burhop; pressure-modulation

effects due to beam-chopping:

Fite) . This is characteristic

of a developing field of

research; old problems surface

in a new context. The

sophistication of the present

research generation is of such

a level that many familiar

problems, seemingly put to

rest, rear their ugly heads

once again.

There is one final point I

would like to address related

to Joachim's talk. He points

out, and correctly, that "it is

usually the cross sections that

are the weakest point" (in

determining the complete set of

observables in a scattering

experiment) . Again, this is a characteristic common to many

branches of physics. Absolute

values of anything are the most

difficult quantities to

determine and this includes but

is not restricted to absolute

cross sections. The field of

coherence and correlation in a certain sense skirts this

problem, since often the

relevant quantities are ratios ; ratios of polarizations, or

scattering signals, for

example. In determining ratios

absolute densities of targets

and absolute intensities of

signals are not required. It is

for this reason that some of

the data obtained herein are

intrinsically more accurate and

reliable than in the supposedly

less demanding integrated or

summed cross sections of more

conventional techniques. Thus

theorists can be presented with

data that they can be

reasonably confident will not

change radically with time.

(Although of course there are

exceptions!) And it must also

be borne in mind that the

complete experiment requires

knowledge of the absolute cross

section, in addition to the

fundamental parameters that are
1 50

the principal focus of this

meeting. My colleague Prof.

Leposava (Lepsha) Vuskovic

addressed this problem in her

talk; I will not elaborate on

it here.

So much for the follow-up on

the Kessler introductory talk.

I would like to make several

further observations that do

not refer to specific

presentations before addressing

some of these. I was struck by

the apparant ubiquitousness of

one particular article that

was referred to in perhaps half

of all the talks presented--

the recent review by Andersen,

Gallagher, and Hertel. Even the

conference logo, showing a laser-excited S to P charge

cloud in an alkali, comes from

that article. Further, although

the article was really intended

as a compendium of experimental

data on polarization and

correlation, the part of the

article that was generally

quoted was only the first few

pages, which summarizes with

great clarity one currently

generally accepted way of

looking at correlation (in

particular Stokes parameters

and reference frames) and

coherence phenomena (L^ and A) , in simple atomic systems. I

suspect that in addition to the

useful presentation of the

relation of observables to

atomic and collision

parameters, a principal reason

for the appeal of this article

is that physicists like to

think pictorially, and it has

some gorgeous sketches that

give at least the illusion that

the excitation process can be

quite readily visualised in

terms of charge clouds,

quantization axes, and

scattering angles. As a corollary to this, now that

this field has reached a certain advanced level in both

theory and experiment, some

conceptually simple unifying

and "obvious" features can now

be recognized (of course these

are simple generally speaking

only after the fact) . An

example is the explanation of

the left-right asymmetry for

polarized electrons scattered

superelastically from laserexcited sodium. See in

particular the discussion in

Sect. 3 of the Hanne article,

and the accompanying Fig. 2.

Now for some specifics. Of the

2 3 research talks 14 were

primarily experimental, 5 were

theoretical, and 4 were devoted

primarily to experimental

"technology" for want of a

better word. It is probably

fair to state that no

revolutionary experimental or

theoretical concepts were

introduced, the meeting being

devoted primarily to the

fruitful exploitation of

methods that are by now

reasonably well-understood.

Certain trends and significant

developments did surface--the

ones I happen to single out do

naturally reflect my own

predelictions rather than some

absolute judgments about what

is particularly important or

not important. The field, which

traditionally concerned itself

primarily with electron-atom

collision has for some time

broadened to include heavy

particle collisions. However,

while 19 of the papers were

dedicated primarily to electron

collisions, including several

that concerned general theory

that could also apply to heavy

particle collisions, there

were only 3 involving atomatom collisions (I'll get to

the 23rd paper later) . It would
151

probably surprise no one were

the ratio of heavy-heavy to

light-heavy particle collisions

to be substantially larger in

Adelaide two years from now.

This development was nicely

highlighted by the two papers,

one theoretical (Masnou-Seeuws)

and the other experimental

(Meijer) , on the associative

ionization reaction in laserexcited Na(3p) + Na(3p). These

papers were concerned with the

observation and analysis of the

dependence of the reaction on

laser polarization which

determines the charge cloud

alignment of the reaction

constituents just as in

electron-atom collisions,

though with greater complexity.

Hippler (Bielefeld) reported

measurements of alignment and

orientation parameters of

atomic hydrogen 2p excitation

either by direct or by chargechanging collisions.

Computational physics was

represented by two impressive

papers from Los Alamos. The

first was a general discussion

of the development of atomic

structure and collision codes

by Clark and the second a talk

more specifically directed at

the use of such codes to

calculate e lectr on- impact

coherence parameters and spinpolarization functions using

first order many body theory

(FOMBT) and distorted wave

approximation schemes (DWBA)

(Cartwright) . It appears that

according to Clark such codes

can now be accessed by modem,

even from NYU. This is an

intriguing development— will

experimentalists be able, at

last, to make serious efforts

at calculating cross sections

and coherence and correlation

parameters? We will doubtless

hear more of this at Adelaide.

Still, the more fundamental

question remains: how reliable

are the actual theoretical

bases for these codes? Among

the liveliest discussions at

the symposium were those

related to the two "schools" of

theory represented, with FOMBT

and DWBA on the one hand, and

close-coupling methods on the

other. These latter were ably

represented by Ian McCarthy

(Flinders) and his US

colleague, Don Madison. The

experimental data at the

symposium produced no welldefined winner. For example.

Prof. Crowe (Belfast)

presented data for P and D

excitation of ground state

helium that did not agree well

with FOMBT and DWBA, but on the

other hand similar data in

krypton did much better. Crowe

suggests that this may be due

to the somewhat smoother

central potential offered to

the exciting electron by the

fatter krypton atom. The

traditional reliability of

close coupling in dealing with

low energy collisions in the

alkalis continued unabated,

with data of the NIST group

(McClelland) on polarized

electron scattering and of the

Flinders group (Teubner) on

orientation and alignment, both

using sodium. At the moment I

would call it a tie, with FOMBT

and DWBA being more successful

(and much more economical with

respect to computer time) at

higher energies, and with close

coupling being more appropriate

as one goes to lower energies.

Also, close coupling may work

better with the alkalis

because, being single valence

electron atoms, their electric

dipole matrix elements tend to

converge relatively rapidly

while the rare gas closed shell

atoms present a smoother
1'52 potential more suited for the

Los Alamos models.

The situation with regard to

spin-tagged electron impact

ionization of atoms seems to be

well in hand, with the

comprehensive measurements of

the Bielefeld group (Baum)

Data for atom.ic hydrogen and

the lighter alkalis ^ as well as

of metastable helium (lithiumlike) were presented. Only a calculation using DWBA by

Bartschat could be compared

with their lithium and

metastable helium data, and

these are in reasonable,

qualitative agreement. The

possibilty of oscillations in

the spin asymmetry close to

threshold seems to have been

put to rest. Prof. McCarthy

pointed out the importance of

establishing the correct

boundary conditions for

ionization reactions, whose

final product contains three

charged bodies. He noted that

Brauner, Briggs and Klar have

recently succeeded in

establishing the exact threebody boundary condition for

this problem. I hope we will be

hearing more about this at the

next symposium.

Inner shell correlation

experiments were reported by

Prof. Stefani (Rome, now at

Camerino) , the only talk that

was devoted primarily to other

than outer-shell excitations.

An interesting experiment was

described by Heideman

(Utrecht), in which he observed

coherence effects in

autoionization of states with

significantly different

excitation energies. Prof»

Stumpf (Moscow, Idaho)

discussed experiments in

progress on optical analysis of

electron excitation of laserexcited alkali atoms. Standage

(Griffith, Australia) , also

employing laser-excited atoms

in related work, described his

measurements of stepwise

excitation. In all I counted

seven papers that used lasers

to prepare, i.e., excite, atoms

(almost exclusively sodium to

3P, excepting barium by JPL) to

perform either superelastic,

elastic (Vuskovic) or inelastic

measurements. There were no

papers that described laserassisted collisions, i.e.,

using high-intensity lasers.

Professor Fano gave a talk

concerned with the dynamics of

the excitation process—the

physics behind the observations

of polarization phenomena. This

is related to the qualitative

discussion by Hanne of the socalled "propensity" describing

the relation between the

orientation of atomic momentum

before collision with respect

to the projectile's angular

momentum, measured with respect

to the atomic nucleus. I should

like to particularly call to

your attention references 5 and

7 of Prof. Fano's article. They

refer to honors B. S. theses by

R. Venugapolan and J. Bohn

respectively. It appears as

though the University of

Chicago continues to attract

first-class undergraduates, not

to mention the fact that it

requires an unusually gifted

mentor to inspire such work

no surprise, in this case!

The twenty third paper, by

Prof. Duncan (Stirling) was not

about collisions at all, being

a study of polarization

correlation effects in the twophoton decay of metastable

deuterium, although by

stretching the point you might

talk yourself into it (the
153

inverse reaction would be a three body (two photon)

collision to excite deuterium

to the 2S state) . This work is

the second paper from Stirling

on the use of atomic systems to

test Bell's inequality, that

is, the validity of quantum

mechanics. Duncan, Kleinpoppen

et al had already performed

such a measurement that

supplemented the original

Aspect et al work. The present

experiment was undertaken to

check out a loophole suggested

by Selleri and coworkers, but

the results still bear out the

predictions of quantum

mechanics. Doubtless other

loopholes will surface in the

future; these will have to be

addressed as they appear. The

point here is that the

techniques being developed to

address the problems of

specific interest to this

symposium may very well be

adapted to other areas of equal

interest and importance.

Finally, to take up the point

raised by Prof. Kessler at the

end of his paper—from what we

have learned in the present

symposium how would I propose

to organize the next one?

Again, I can't rally the

chutzpa (read, presumption) to

offer such an outline--it is

better to just wait and see

what develops. I certainly

would not be surprised to see

more papers on heavy particle

collisions, certainly more

papers involving laser-prepared

collision targets, and maybe

even some work on laserassisted collisions, and

perhaps a paper or two

involving exotic collision

partners, particularly

positrons. Very likely still

more accurate measurement of

scattering amplitudes either

through the Lj_ , 7 or A , X

approaches will be forthcoming,

to approach ever closer to the

"perfect scattering

experiment" . In this connection

it should be noted that one can

go even beyond this ideal, in

si m. pie electron-atom

collisions, because of the possibility of

overdetermination of collision

parameters through the

performance of more than the

required number of independant

experiments. (Purists can argue

that if the parameters are

overdetermined the experiments

cannot be all independent. This

is true only if timeindependant scattering theory

as we understand it represents

a completed description of the

scattering process. Future

precision experiments will play

a role in deciding, in fact, if

this is actually the case.) I

would most assuredly like to

see a more elaborate face-off

arranged with a larger group of

theorists to discuss not only inevitably improved

computational techniques, but

more fundamental theoretical

talks as well. And finally it

is always important when

experts meet to address the

highly specialized but crucial

technical questions that

determine in the last analysis

the reliability of experimental

results, as was noted at the

begining of this paper. I hope

that there will be more of this

at Adelaide, although perhaps

we don't have to force our

theoretical colleagues to sit

through some of the more nittygritty discussion that will

inevitably occur.
I

i
155

List of Attendees

J. Abdallah Los Alamos National Laboratory, USA

N. Andersen University of Aarhus, Denmark

K. Bartschat Drake University, USA

W. Batelaan University of Utrecht, The Netherlands

G. Baum University of Bielefeld, FRG

K. Becker City College of CUNY, USA

B. Bederson New York University, USA

J. A. Brand University of Missouri, Rolla, USA

S. Buckman Australia National University

L. Bureyeva USSR Academy of Sciences, USSR

D.C. Cartwright Los Alamos National Laboratory, USA

R.E.H. Clark Los Alamos National Laboratory, USA

J.J. Corr University of Windsor, Canada

A. Crowe (jueen s University oi Belfast, UK

D.M. Crowe City College of CUNY, USA

G. Csanak X \1 TVTi' 1X1 J XT04 Los Alamos National Laboratory, USA

A.J. Duncan XT' ' i C O i 1 ' X T X University oi Sterling, UK

U. Fano XT • 'i C /"n * XTO A University oi Chicago, USA

T.J. Gay XT* * J /T ' " X*^ 11 XT A University oi Missouri, Kolia, USA

X.Q. Guo City College of CUNY, USA

/"I "n TT G.F. Hanne XT* * 1 {* Ti if i X^ T~\ /"'i University oi Munster, rKG

H.G.M. Heideman University of Utrecht, The Netherlands

R. Hippler University of Bielefeld, FRG

K-N. Huang Institute of Atomic and Molecular

Science, Taiwan

A. Jain Florida State University, USA

T.Y. Jiang New York University, USA

M.H. Kelley N.LS.T, Gaithersburg, USA

J. Kessler University of Munster, FRG

M.A. Khakoo University of Missouri, Rolla, USA

H. Kleinpoppen University of Sterling, UK

C.J. Liu Argonne National Laboratory, USA

M.S. Lubell City College of CUNY, USA

W.R. MacGillivray Griffith University, Australia

L.E. Machado University of Sao Carlos, Brazil

D.H. Madison University of Missouri, Rolla, USA
156

T-l T » /T 1 TT b.J. Mansky li ueorgia iecn., UbA

B. Marinkovic Institute of Physics, Belgrade,

Yugoslavia

Iv.jiv. iviartus v^ity vvOiiege oi v^uin i , uo/\

F. Masnou-Seeuws Universite de Paris-Sud, France

IV. u. iviainur university oi xvoorKee, lUQia

i. ivicv^arcny riiiiu.eib uiiiveibiiy, rvubiiaiid. J.J. iVicL'leiiana iN.i.o.i, LiaitnersDurg, uoA

J . vv . ivicL^oiiivey universii}'^ OI vvinQsor, v^anaaa

sx. iViCEjacnraii York University, Canada

ii.A.j. ivieijer University of Kaiserslautern, FRG

U . Muller University of Saarlandes, FRG

r.A. JNeiil University of Nevada, Reno, USA

O T7 "NT* 1 b.ti. JNielsen University oi Copenhagen, Denmark

J. Katlm T J- T) 1 T 1, 4- TTC A Jet Propulsion Laboratory, UbA

b. Keicnert University oi Mamz, hRG

TT T~> O J 1- H.K,. badeghpour TTT A TT 'L r 1 1 TTC A J iL A- University oi Colorado, UbA

b. barkar Indian Association tor the Cultivation

OI bcience, India

K. biegel r~^4+,T ^^^^^ /"'tttvtv ttc a City College oi CUiN Y, UbA

M.Kj. btandage Griffith University, Australia

Kj. bteiani Universita' di Camerino, Italy

xl. bteidl University oi Bieleield, rrCC

B.J. Stumpf University of Idaho, Moscow, USA

r.J.U. ieubner X?^ rlinders 'J TT' University, "J-Aj-T Australia

b. i raj mar TTDT I, TTC A JrL-Caltecn, UbA

IT} J ID 1 J.r. van den Brink TT * '

i. r TTJ- "Li- f~f^1^ AT il- 1 J University OI Utrecht, ihe JNetheriands

r.J.M. van der rJurgt University of Windsor, Canada

L. Vuskovic IN I u, uoA/ msiitute oi rnysics, oeigra.

Yugoslavia

K. Wijayaratna University of Missouri, Rolla, USA

J.F. Williams University of Western Australia

P.W.. Zetner JPL-Caltech, USA

Y. Zheng Flinders University, Australia
NIST-114A

(REV. 3-90) U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

1. PUBLICATION OR REPORT NUMBER

kftc;t/c;p-7P!Q

2. PERFORMING ORGANIZATION REPORT NUMBER

BIBLIOGRAPHIC DATA SHEET 3. PUBUCATION DATE June 1990

4. TITLE AND SUBTITLE Proceedings of the International Symposium on Correlation and Polarization in

Electronic and Atomic Collisions

5. AUTHOR(S)

P. A. Neill, K.H. Becker and M.H. Kelley

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

QAITHERSBURQ, MO 20S99

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP) Electron Physics Group

Electron and Optical Physics Division

Building 220, Room B206

N.I.S.T., Gaithersburg, Maryland 20899

10. SUPPLEMENTARY NOTES

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

LITERATURE SURVEY, MENTION IT HERE.)

The International Symposium on Correlation and Polarization in Electronic

and Atomic Collisions was held at the Stevens Institute of Technology, Hoboken,

New Jersey on August 2-4, 1989. The previous meeting, held in Belfast, UK (1987),

extended the scope of the Symposium to include ion-atom and atom-atom collisions.

This was reflected in the Symposium title which was modified to replace the term

"electron-atom collisions" with "electronic and atomic collisions." The Hoboken

meeting maintained this expanded area and included several talks in more loosely

related areas. The attendance of 66 scientists represented 16 countries and the

talks described the work of 23 research groups from 9 countries.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

alignment; coherence; correlation; electronic and atomic collisions; electron-photon

coincidence; orientation; polarization; scattering

13. AVAILABIUTY UNUNITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

X ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE, WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

163

15. PRICE

ELECTRONIC FORM

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NIST.Technical Publications

Periodical

Journal of Research of the National Institute of Standards and Technology—Reports NIST research

and development in those disciplines of the physical and engineering sciences in which the Institute

is active. These include physics, chemistry, engineering, mathematics, and computer sciences.

Papers cover a broad range of subjects, with major emphasis on measurement methodology and

the basic technology underlying standardization. Also included from time to time are survey articles on topics closely related to the Institute's technical and scientific programs. Issued six times a year.

Nonperiodicals

Monographs—Major contributions to the technical literature on various subjects related to the

Institute's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies.

Special Publications—Include proceedings of conferences sponsored by NIST, NIST annual reports,

and other special publications appropriate to this grouping such as wall charts, pocket cards, and

bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in

scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical

properties of materials, compiled from the world's literature and critically evaluated. Developed un- der a worldwide program coordinated by NIST under the authority of the National Standard Data

Act (Pubhc Law 90-396). NOTE: The Journal of Physical and Chemical Reference Data (JPCRD)

is published quarterly for NIST by the American Chemical Society (ACS) and the American Insti- tute of Physics (AIP). Subscriptions, reprints, and supplements are available from ACS, 1155 Sixteenth St., NW., Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Institute on building

materials, components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental functions and the

durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treat- ment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NIST

under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationally

recognized requirements for products, and provide all concerned interests with a basis for common

understanding of the characteristics of the products. NIST administers this program as a supplement

to the activities of the private sector standardizing organizations.

Consumer Information Series—Practical information, based on NIST research and experience, covering areas of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order the above NISTpublications from: Superintendent of Documents, Government Printing Office,

Washington. DC 20402.

Order the following NISTpublications—FIPS and NISTIRs—from the National Technical Information

Service, Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series col- lectively constitute the Federal Information Processing Standards Register. The Register serves as the official source of information in the Federal Government regarding standards issued by NIST

pursuant to the Federal Property and Administrative Services Act of 1949 as amended. Public Law

89-306 (79 Stat. 1127), and as implemented by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NIST Interagency Reports (NISTIR)—A special series of interim or final reports on work performed

by NIST for outside sponsors (both government and non-government). In general, initial distribu- tion is handled by the sponsor; public distribution is by the National Technical Information Service,

Springfield, VA 22161, in paper copy or microfiche form.
U.S. Department of Commerce

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(formeriy National Bureau of Standards)

Gaitherslxirg, MD 20899

Official Business

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